Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

Abstract

Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in [0, h_{\mathrm {top}}(f))$, there exists an ergodic measure $\mu $ of f satisfying $h_{\mu }(f)=c$. In this paper, we obtain a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for basic sets of flows that characterize the refined roles of ergodic measures in the invariant ones. In this process, we establish a ‘multi-horseshoe’ entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain the same result for singular hyperbolic attractors.

1 Introduction

In [22], Katok showed that every $C^{1+\alpha }$ diffeomorphism f in dimension $2$ has horseshoes of large entropies. This implies that the system has the intermediate entropy property, that is, for any constant $c \in [0, h_{\mathrm {top}}(f))$ , there exists an ergodic measure $\mu $ of f satisfying $h_{\mu }(f)=c,$ where $h_{\mathrm {top}}(f)$ is the topological entropy of f and $h_{\mu }(f)$ is the metric entropy of $\mu .$ Katok believed that this holds in any dimension. In the last decade, a number of partial results on the intermediate entropy property have been obtained, see [10, 17, 21, 23, 24, 26, 27, 32, 35, 37].

Conjecture 1.1. (Katok)

Every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property.

In [36], the authors consider the intermediate Birkhoff average. To be precise, they proved that if $f: X \rightarrow X$ is a continuous map over a compact metric space X with the periodic gluing orbit property, then there is an ergodic measure $\mu _{\alpha } \in \mathcal {M}(f, X)$ such that $\int g\,d\mu _{\alpha }=\alpha $ for any continuous function $g:X\to \mathbb {R}$ and any constant $\alpha $ satisfying

$$ \begin{align*} \inf _{\mu \in \mathcal{M}(f, X)} \int g\,d\mu<\alpha<\sup _{\mu \in \mathcal{M}(f, X)} \int g\,d\mu, \end{align*} $$

where $\mathcal {M}(f, X)$ is the set of all f-invariant probability measures. (The author obtained the same result for the asymptotically additive functions, readers can refer to [36] for details.)

Imitating the relationships between extremum and conditional extremum, variational principle, and conditional variational principle, we consider the following question.

Question 1.1. Under certain restricted conditions, do the intermediate entropy property and intermediate Birkhoff average property hold?

In fact, a certain restricted condition determines a subset of $\mathcal {M}(f, X).$ So in other words, Question 1.1 means that given a subset F of $\mathcal {M}(f, X),$ does one have

(1.1) $$ \begin{align} \{h_{\mu}(f):\mu\in F\}=\{h_{\mu}(f):\mu\in F\cap \mathcal{M}_{\mathrm{erg}}(f,X)\} \end{align} $$

and

(1.2) $$ \begin{align} \bigg\{\!\int g\,d\mu:\mu\in F\bigg\}=\bigg\{\!\int g\,d\mu:\mu\in F\cap \mathcal{M}_{\mathrm{erg}}(f,X)\bigg\}? \end{align} $$

Here $\mathcal {M}_{\mathrm {erg}}(f,X)$ is the set of all f-invariant ergodic probability measures.

It is clear that Question 1.1 can only be true under some suitable restricted condition or some suitable $F.$ For example, if $F\cap \mathcal {M}_{\mathrm {erg}}(f,X)=\emptyset ,$ then Question 1.1 is obviously false. When $F=\mathcal {M}(f,X),$ equation (1.1) is true for every $C^{1+\alpha }$ diffeomorphism f in dimension $2$ [22] and equation (1.2) is true for a system satisfying the periodic gluing orbit property [36]. In the two proofs, a basic requirement is that $\mathcal {M}_{\mathrm {erg}}(f,X)$ is dense in $\mathcal {M}(f,X).$ So we believe that a suitable F should satisfy the following properties:

1. (1) $\mathcal {M}_{\mathrm {erg}}(f,X)\cap F$ is dense in $F;$

2. (2) F is convex.

We will see that the following two sets satisfy items (1) and (2) for hyperbolic sets:

$$ \begin{align*} F_1(c)=\{\mu\in \mathcal{M}(f,X):h_{\mu}(f)=c\},\quad\ F_2^g(\alpha)=\bigg\{\mu\in \mathcal{M}(f,X):\int g\,d\mu=\alpha\bigg\}, \end{align*} $$

where $c \in [0, h_{\mathrm {top}}(f))$ and $ \inf _{\mu \in \mathcal {M}(f, X)} \int g\, d\mu <\alpha <\sup _{\mu \in \mathcal {M}(f, X)} \int g\,d\mu. $ Additionally, for two continuous functions $g,h$ , we will consider whether the following three equalities hold: a conditional intermediate entropy property

(1.3) $$ \begin{align} \{h_{\mu}(f):\mu\in F_2^g(\alpha)\}=\{h_{\mu}(f):\mu\in F_2^g(\alpha)\cap \mathcal{M}_{\mathrm{erg}}(f,X)\}, \end{align} $$

and two conditional intermediate Birkhoff average properties

(1.4) $$ \begin{align} \bigg\{\!\int g\,d\mu:\mu\in F_1(c)\bigg\}=\bigg\{\!\int g\,d\mu:\mu\in F_1(c)\cap \mathcal{M}_{\mathrm{erg}}(f,X)\bigg\}, \end{align} $$

(1.5) $$ \begin{align} \bigg\{\!\int h\,d\mu:\mu\in F_2^g(\alpha)\bigg\}=\bigg\{\!\int h\,d\mu:\mu\in F_2^g(\alpha)\cap \mathcal{M}_{\mathrm{erg}}(f,X)\bigg\}. \end{align} $$

Note that if the following two sets are equal,

$$ \begin{align*} \begin{aligned} &\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(f)\bigg):\mu\in \mathcal{M}(f,X)\bigg\},\\ &\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(f)\bigg):\mu\in \mathcal{M}_{\mathrm{erg}}(f,X)\bigg\}, \end{aligned} \end{align*} $$

then equations (1.3) and (1.4) hold except extremums for any $c \in [0, h_{\mathrm {top}}(f))$ and $ \inf _{\mu \in \mathcal {M}(f, X)}\!\int \! g\,d\mu <\alpha <\sup _{\mu \in \mathcal {M}(f, X)} \int g\,d\mu. $ Additionally, if the following two sets are equal,

$$ \begin{align*} \begin{aligned} &\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, \int h\,d\mu\bigg):\mu\in \mathcal{M}(f,X)\bigg\},\\ &\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, \int h\,d\mu\bigg):\mu\in \mathcal{M}_{\mathrm{erg}}(f,X)\bigg\}, \end{aligned} \end{align*} $$

then equation (1.5) holds except extremums for any $ \inf _{\mu \in \mathcal {M}(f, X)} \int g\,d\mu <\alpha <\sup _{\mu \in \mathcal {M}(f, X)} \int g\,d\mu. $

In this paper, we are interested in flows. We first recall some notions of flows. Let $\mathscr {X}^r(M)$ , $r\ge 1$ , denote the space of $C^r$ -vector fields on a compact Riemannian manifold M endowed with the $C^r$ topology. For $X\in \mathscr {X}^r(M)$ , denote by $\phi _t^X$ or $\phi _t$ for simplicity the $C^r$ -flow generated by X and denote by $\mathrm {D}\phi _t$ the tangent map of $\phi _t$ . Given a vector field $X\in \mathcal {X}^1(M)$ and a compact invariant subset $\Lambda $ of the $C^1$ -flow $\Phi =(\phi _t)_{t\in \mathbb {R}}$ generated by X, we denote by $C(\Lambda ,\mathbb {R})$ the space of continuous functions on $\Lambda $ . The set of invariant (respectively ergodic) probability measures of X supported on $\Lambda $ is denoted by $\mathcal {M}(\Phi ,\Lambda )$ (respectively $\mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda )$ ), and it is endowed with the weak $^*$ -topology. We denote by $h_{\mu }(X)$ or $h_{\mu }(\Phi )$ the metric entropy of the invariant probability measure $\mu \in \mathcal {M}(\Phi ,\Lambda )$ , defined as the metric entropy of $\mu $ with respect to the time-1 map $\phi ^X_1$ of the flow. Let $d^*$ be a translation invariant metric on the space $\mathcal {M}(\Phi ,\Lambda )$ compatible with the weak $^*$ topology. We focus on the following question for flows in the present paper.

Question 1.2. For every flow $(\phi _t)_{t\in \mathbb {R}}$ generated by a typical vector field X, and two continuous functions g, h on a compact invariant subset $\Lambda $ of the $C^1$ flow $(\phi _t)_{t\in \mathbb {R}},$ do the following two equalities hold:

$$ \begin{align*} \mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(\Phi)\bigg):\mu\in \mathcal{M}(\Phi,\Lambda)\bigg\} =\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(\Phi)\bigg):\mu\in \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\bigg\} \end{align*} $$

and

$$ \begin{align*} \mathrm{Int}\bigg\{\bigg(\!\int g\,d\mu, \int h\,d\mu\bigg):\mu\!\in\! \mathcal{M}\bigg(\Phi,\Lambda\bigg)\bigg\}\! =\!\mathrm{Int}\bigg\{\bigg(\int g\,d\mu, \int h\,d\mu\bigg):\mu\in \mathcal{M}_{\mathrm{erg}}\bigg(\Phi,\Lambda\bigg)\bigg\}? \end{align*} $$

We will answer this question partially for hyperbolic flows and singular hyperbolic flows.

Definition 1.3. Given $X\in \mathscr {X}^1(M)$ , an invariant compact set $\Lambda $ is called a basic set if it is transitive, hyperbolic, and locally maximal; is not reduced to a single orbit of a hyperbolic critical element; and its intersection with any local cross-section to the flow is totally disconnected.

Given a continuous function g on a compact invariant subset $\Lambda $ of the $C^1$ flow $(\phi _t)_{t\in \mathbb {R}},$ denote $L_{g}=\{ \int g\,d\mu :\mu \in \mathcal {M}(\Phi ,\Lambda )\}.$ For any $\alpha \in L_{g},$ denote

$$ \begin{align*} M_{g}(\alpha)=\bigg\{\mu\in \mathcal{M}(\Phi,\Lambda):\int g\,d\mu=\alpha\bigg\},\quad\ M_{g}^{\mathrm{erg}}(\alpha)=M_{g}(\alpha)\cap \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\end{align*} $$

and

$$ \begin{align*}M_{g}^{\mathrm{top}}(\alpha)=\{\mu\in M_g(\alpha):h_{\mu}(\Phi)=\sup\{h_{\nu}(\Phi):\nu\in M_{g}(\alpha)\}\}.\end{align*} $$

Then $M_{g}(\alpha )$ is a closed subset of $\mathcal {M}(\Phi ,\Lambda )$ and thus

$$ \begin{align*}\sup\{h_{\nu}(\Phi):\nu\in M_{g}(\alpha)\}=\max\{h_{\nu}(\Phi):\nu\in M_{g}(\alpha)\}\end{align*} $$

for any $\alpha \in L_{g}$ if the entropy function $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. For a probability measure $\mu \in \mathcal {M}(\Phi ,\Lambda )$ , we denote the support of $\mu $ by

$$ \begin{align*}S_{\mu}:=\{x\in \Lambda:\mu(U)>0\ \text{for any neighborhood}\ U\ \text{of}\ x\}.\end{align*} $$

Now we state our first main result.

Theorem A. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set of X. If g is a continuous function on $\Lambda $ , then for any $\alpha \in \mathrm {Int}(L_{g}),$ any $\mu \in M_{g}(\alpha )\setminus M_{g}^{\mathrm {top}}(\alpha ),$ any $0\leq c\leq h_{\mu }(\Phi )$ , and any $\zeta>0$ , there is $\nu \in M_{g}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu )<\zeta $ and $h_{\nu }(\Phi )=c.$ Moreover, for any $\alpha \in \mathrm {Int}(L_{g})$ and $0\leq c< \max \{h_{\mu }(\Phi ):\mu \in M_{g}(\alpha )\},$ the set $\{\mu \in M_{g}^{\mathrm {erg}}(\alpha ):h_{\mu }(\Phi )=c, S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{g}(\alpha ):h_{\mu }(\Phi )\geq c\}.$

Remark 1.4. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set of X. For a continuous function g, let $g_{t}=\int _{0}^{t} g(\phi _{\tau }(x))\,d\tau ,$ then $(g_t)_{t\geq 0}$ is an additive family of continuous functions. So let $\chi \equiv 0$ and $d=1$ in Theorem 4.1(iv), we have

(1.6) $$ \begin{align} \mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(\Phi)\bigg):\mu\in\mathcal{M}(\Phi,\Lambda)\bigg\}= \mathrm{Int}\bigg\{\bigg(\int g\,d\mu, h_{\mu}(\Phi)\bigg):\mu\in \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\bigg\}. \end{align} $$

We draw the graph of $\big(\int g\,d\mu , h_{\mu }(\Phi )\big)$ in Figure 1. Then by equation (1.6), every point in the interior of the region can be attained by ergodic measures. From equation (1.6), we obtain one conditional intermediate metric entropy property:

$$ \begin{align*}&\mathrm{Int}\bigg\{h_{\mu}(\Phi):\mu\in \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\text{ and }\int g\,d\mu=\alpha\bigg\}\\&\quad=\mathrm{Int}\bigg\{h_{\mu}(\Phi):\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }\int g\,d\mu=\alpha\bigg\},\end{align*} $$

and one conditional intermediate Birkhoff average property:

$$ \begin{align*} &\mathrm{Int}\bigg\{\!\int g\,d\mu:\mu\in \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\text{ and }h_{\mu}(\Phi)=c\bigg\}\\ &\quad=\mathrm{Int}\bigg\{\!\int g\,d\mu:\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }h_{\mu}(\Phi)=c\bigg\}\end{align*} $$

for any $\alpha \in \mathrm {Int}(L_{g})$ and any $c\in (0,h_{\mathrm {top}}(\Lambda )).$

Figure 1 Graph of $\big(\int g\,d\mu , h_{\mu }(\Phi )\big)$ .

Remark 1.5. Given $X \in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , $\mathcal {M}(\Phi ,\Lambda )$ is said to have entropy-dense property if for any $\varepsilon>0$ and any $\mu \in \mathcal {M}(\Phi ,\Lambda )$ , there exists $v \in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda )$ satisfying

$$ \begin{align*} d^*(\mu, v)<\varepsilon \quad\text{and}\quad h_{\nu}(\Phi)>h_{\mu}(\Phi)-\varepsilon. \end{align*} $$

Given a continuous function g on $\Lambda ,$ we say $\mathcal {M}(\Phi ,\Lambda )$ has entropy-dense property with the same g-level if for any $\alpha \in \mathrm {Int}(L_{g}),$ any $\mu \in M_{g}(\alpha )$ , and any $\varepsilon>0$ , there is $\nu \in M_{g}^{\mathrm {erg}}(\alpha )$ such that

$$ \begin{align*} d^*(\mu, v)<\varepsilon \quad\text{and}\quad h_{\nu}(\Phi)>h_{\mu}(\Phi)-\varepsilon. \end{align*} $$

By Theorem A, if $\Lambda $ is a basic set of X and g is a continuous function on $\Lambda $ , then $\mathcal {M}(\Phi ,\Lambda )$ has entropy-dense property with the same g-level. Moreover, we can choose $\nu \in M_{g}^{\mathrm {erg}}(\alpha )$ such that $d^*(\mu , v)<\varepsilon $ and $ h_{\nu }(\Phi )=h_{\mu }(\Phi ) $ if $\mu \in M_{g}(\alpha )\setminus M_{g}^{\mathrm {top}}(\alpha ).$

Given two continuous functions g, h on a compact invariant subset $\Lambda $ of the $C^1$ flow $(\phi _t)_{t\in \mathbb {R}},$ we denote

$$ \begin{align*} L_{g,h}=\bigg\{\bigg(\int g\,d\mu, \int h\,d\mu\bigg):\mu\in \mathcal{M}(\Phi,\Lambda)\bigg\}. \end{align*} $$

Then $L_{g,h}$ is a non-empty convex compact subset of $\mathbb {R}^2.$ For any $\alpha \in L_{g,h},$ denote

$$ \begin{align*} M_{g,h}(\alpha)&=\bigg\{\mu\in \mathcal{M}(\Phi,\Lambda):\bigg(\int g\,d\mu, \int h\,d\mu\bigg)=\alpha\bigg\},\ M_{g,h}^{\mathrm{erg}}(\alpha)\\&=M_{g,h}(\alpha)\cap \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda). \end{align*} $$

Now we state our second main result.

Theorem B. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set of X. If $g,h$ are continuous functions on $\Lambda $ , then for any $\alpha \in \mathrm {Int}(L_{g,h}),$ any $\mu \in M_{g,h}(\alpha )$ , and any $\zeta>0$ , there is $\nu \in M_{g,h}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu )<\zeta .$ Moreover, for any $\alpha \in \mathrm {Int}(L_{g,h}),$ the set $\{\mu \in M_{g,h}^{\mathrm {erg}}(\alpha ):S_{\mu }=\Lambda \}$ is residual in $M_{g,h}(\alpha ).$

Remark 1.6. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set of X. If $g,h$ are continuous functions on $\Lambda $ , then by Theorem B, we have

$$ \begin{align*}\mathrm{Int}\bigg\{\bigg(\!\int g\,d\mu, \int h\,d\mu\bigg):\mu\in \mathcal{M}(\Phi,\Lambda)\bigg\} =\mathrm{Int}\bigg\{\bigg(\!\int g\,d\mu, \int h\,d\mu\bigg):\mu\!\in\! \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\bigg\}.\end{align*} $$

Then we obtain a conditional intermediate Birkhoff average property:

$$ \begin{align*}&\mathrm{Int}\bigg\{\!\int h\,d\mu:\mu\in \mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\text{ and }\int g\,d\mu=\alpha\bigg\}\\&\quad=\mathrm{Int}\bigg\{\!\int h\,d\mu:\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }\int g\,d\mu=\alpha\bigg\}\end{align*} $$

for any $\alpha \in \mathrm {Int}(L_{g}).$

In the process of proving Theorems A and B, there are two keypoints: ‘multi-horseshoe’ entropy-dense property (see Theorem 2.5) and conditional variational principles (see Theorems 3.5 and 3.7) proved by Barreira and Holanda in [5, 20].

1.1 Outline of the paper

In §2, we introduce ‘multi-horseshoe’ entropy-dense property for flows and prove that it holds for basic sets. In §3, we recall non-additive thermodynamic formalism for flows, give abstract conditions on which the results of Theorems A and B hold in the more general context of asymptotically additive families of continuous functions. In §4, using the ‘multi-horseshoe’ entropy-dense property, we show that the abstract conditions given in §3 are satisfied for basic sets, and thus we obtain Theorems A and B. In §5, we consider singular hyperbolic attractors and give corresponding results on Question 1.2.

2 ‘Multi-horseshoe’ entropy-dense property

In this section, we introduce the ‘multi-horseshoe’ entropy-dense property and prove it holds for basic sets. We first recall the definition of hyperbolicity.

Definition 2.1. Given a vector field $X\in \mathscr {X}^1(M)$ , a compact $\phi _t$ -invariant set $\Lambda $ is hyperbolic if $\Lambda $ admits a continuous $\mathrm {D}\phi _t$ -invariant splitting $T_{\Lambda }M=E^s\oplus \langle X\rangle \oplus E^u$ , where $\langle X\rangle $ denotes the one-dimensional linear space generated by the vector field, and $E^s$ (respectively $E^u$ ) is uniformly contracted (respectively expanded) by $\mathrm {D}\phi _t$ , that is to say, there exist constants $C>0$ and $\eta>0$ such that for any $x\in \Lambda $ and any $t\geq 0$ :

• • $\|\mathrm {D}\phi _t(v)\|\leq Ce^{-\eta t}\|v\|$ for any $v\in E^s(x)$ ; and

• • $\|\mathrm {D}\phi _{-t}(v)\|\leq Ce^{-\eta t}\|v\|$ for any $v\in E^u(x)$ ,

for any $x\in \Lambda $ and $t\geq 0$ . A hyperbolic set $\Lambda $ is said to be locally maximal if there exists an open neighborhood U of $\Lambda $ such that $\Lambda =\bigcap _{t\in {\mathbb R}}\phi _t(U)$ ).

Now we give the definition of a horseshoe.

Definition 2.2. Given $X\in \mathscr {X}^1(M)$ and a hyperbolic invariant compact set $\Lambda $ , we call $\Lambda $ a horseshoe of $\phi _{t}$ if there exists a suspension flow $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ over a transitive subshift of finite type $(\Delta ,\sigma )$ with a Lipschitz roof function $\rho $ and a homeomorphism $\pi : \Delta _{\rho } \rightarrow \Lambda $ such that $\pi \circ f_t=\phi _t\circ \pi .$

Remark 2.3. Lipschitz continuity of $\rho $ is used in Lemma 3.1.

For any $m\in {\mathbb {N}}$ and $\{\nu _i\}_{i=1}^m \subseteq \mathcal {M}(\Phi ,\Lambda )$ , we denote the convex combination of $\{\nu _i\}_{i=1}^m$ by

$$ \begin{align*} \operatorname{\mathrm{cov}}\{\nu_i\}_{i=1}^m:=\bigg\{\sum_{i=1}^m\theta_i\nu_i:\theta_i\in[0, 1], 1\leq i\leq m~\textrm{and}~\sum_{i=1}^m\theta_i=1\bigg\}. \end{align*} $$

For any non-empty subsets of $\mathcal {M}(\Phi ,\Lambda )$ , A and $B,$ we denote the Hausdorff distance between them by

$$ \begin{align*} d_H(A, B):=\max\Big\{\sup_{x\in A}\inf_{y\in B}d^*(x, y),\sup_{y\in B}\inf_{x\in A}d^*(y, x) \Big\}. \end{align*} $$

We denote by $h_{\mathrm {top}}(\Lambda )$ the topological entropy of $\Lambda $ , defined as the topological entropy of $\Lambda $ with respect to the time-1 map $\phi _1$ of the flow.

Definition 2.4. Given $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , let $\mathcal {N}\subset \mathcal {M}(\Phi ,\Lambda )$ be a non-empty set. We say $\Lambda $ satisfies the ‘multi-horseshoe’ entropy-dense property on $\mathcal {N}$ (abbreviated ‘multi-horseshoe’ dense property) if for any $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {N}$ and any $\eta , \zeta>0$ , there exist compact invariant subsets $\Lambda _i\subseteq \Theta \subsetneq \Lambda $ such that for each $1\leq i\leq m$ :

1. (1) $\Lambda _i$ and $\Theta $ are horseshoes;

2. (2) $h_{\mathrm {top}}(\Lambda _i)>h_{\mu _i}(X)-\eta $ and consequently, $h_{\mathrm {top}}(\Theta )>\sup \{h_{\kappa }(X):\kappa \in F\}-\eta ;$

3. (3) $d_H(F, \mathcal {M}(\Phi ,\Theta ))<\zeta $ , $d_H(\mu _i, \mathcal {M}(\Phi ,\Lambda _i))<\zeta $ .

For convenience, when $\mathcal {N}=\mathcal {M}(\Phi ,\Lambda )$ , we say $\Lambda $ satisfies the ‘multi-horseshoe’ entropy-dense property.

Theorem 2.5. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set of X. Then $\Lambda $ satisfies the ‘multi-horseshoe’ dense property.

We will prove Theorem 2.5 by three steps. In the process, we give various versions of ther ‘multi-horseshoe’ dense property on transitive subshifts of finite type, suspension flows over transitive subshifts of finite type, and basic sets.

2.1 Transitive subshifts of finite type

For a homeomorphism $f:K\to K$ of a compact metric space $(K,d)$ , we denote by $\mathcal {M}(f,K)$ the space of f-invariant probability measures. Additionally, let $h_{\mathrm {top}}(f,\Lambda )$ denote the topological entropy of an f-invariant compact set $\Lambda $ and $h_{\mu }(f)$ denote the metric entropy of an f-invariant measure. In [15], a result on the ‘multi-horseshoe’ dense property of homeomorphisms was obtained.

Theorem 2.6. [15, Theorem 2.6]

Suppose a homeomorphism $f:K\to K$ of a compact metric space $(K,d)$ is transitive, expansive, and satisfies the shadowing property. Then for any $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(f, K),$ any $x\in K$ , and any $\eta , \zeta ,\varepsilon>0$ , there exist compact invariant subsets $\Lambda _i\subseteq \Xi \subsetneq K$ such that for each $1\leq i\leq m$ :

1. (1) $(\Lambda _i,f|_{\Lambda _i})$ and $(\Xi ,f|_{\Xi })$ are transitive, expansive, and satisfy the shadowing property;

2. (2) $h_{\mathrm {top}}(f, \Lambda _i){\kern-1pt}>{\kern-1pt}h_{\mu _i}(f){\kern-1pt}-{\kern-1pt}\eta $ and, consequently, $h_{\mathrm {top}}(f, \Xi ){\kern-1pt}>{\kern-1pt}\sup \{h_{\kappa }(f):\kappa {\kern-1pt}\in{\kern-1pt} F\}-\eta ;$

3. (3) $d_H(F, \mathcal {M}(f, \Xi ))<\zeta $ , $d_H(\mu _i, \mathcal {M}(f, \Lambda _i))<\zeta $ ;

4. (4) there is a positive integer L such that for any $z\in \Xi $ , one has $d(f^{j+mL}(z),x)<\varepsilon $ for some $0\leq j\leq L-1$ and any $m\in \mathbb {Z}$ .

Let k be a positive integer. Consider the two-side full symbolic space

$$ \begin{align*} \Sigma=\prod_{-\infty}^{\infty} \{0, 1, \ldots, k-1\}, \end{align*} $$

and the shift homeomorphism $\sigma : \Sigma \to \Sigma $ defined by

$$ \begin{align*} (\sigma(w))_{n}=w_{n+1}, \end{align*} $$

where $w =(w_{n})_{-\infty }^{\infty }.$ A metric on $\Sigma $ is defined by $d(x, y)=2^{-m}$ if m is the largest positive integer with $x_{n}=y_{n}$ for any $|n|<m$ , and $d(x, y)=1$ if $x_{0} \neq y_{0}.$ If $\Delta $ is a closed subset of $\Sigma $ with $\sigma (\Delta )=\Delta ,$ then $\sigma |_{\Delta }: \Lambda \to \Delta $ is called a subshift. We usually write this as $\sigma : \Delta \to \Delta .$ A subshift $\sigma : \Delta \rightarrow \Delta $ is said to be of finite type if there exists some positive integer N and a collection of blocks of length $N+1$ with the property that $x=(x_{n})_{-\infty }^{\infty } \in \Delta $ if and only if each block $(x_{i}, \ldots , x_{i+N})$ in x of length $N+1$ is one of the prescribed blocks.

Recall from [38] a subshift satisfies the shadowing property if and only if it is a subshift of finite type. As a subsystem of two-side full shift, it is expansive. So we have the following corollary.

Corollary 2.7. Suppose $\sigma :\Delta \to \Delta $ is a transitive subshift of finite type. Then for any $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\sigma , \Delta ),$ any $x\in \Delta $ , and any $\eta , \zeta ,\varepsilon>0$ , there exist compact invariant subsets $\Delta _i\subseteq \Xi \subsetneq \Delta $ such that for each $1\leq i\leq m$ :

1. (1) $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ are transitive subshifts of finite type.

2. (2) $h_{\mathrm {top}}(\sigma , \Delta _i){\kern-1pt}>{\kern-1pt}h_{\mu _i}(\sigma )-\eta $ and consequently, $h_{\mathrm {top}}(\sigma , \Xi ){\kern-1pt}>{\kern-1pt}\sup \{h_{\kappa }(\sigma ):\kappa {\kern-1pt}\in{\kern-1pt} F\}-\eta ;$

3. (3) $d_H(F, \mathcal {M}(\sigma , \Xi ))<\zeta $ , $d_H(\mu _i, \mathcal {M}(\sigma , \Delta _i))<\zeta $ ;

4. (4) there is a positive integer L such that for any $z\in \Xi $ , one has $d(\sigma ^{j+mL}(z),x)<\varepsilon $ for some $0\leq j\leq L-1$ and any $m\in \mathbb {Z}$ .

2.2 Suspension flows over transitive subshifts of finite type

Let $f\colon K\to K$ be a homeomorphism on a compact metric space $(K,d)$ and consider a continuous roof function $\rho \colon K\to (0,+\infty )$ . We define the suspension space to be

$$ \begin{align*}K_{\rho}=\{(x,s)\in K\times [0,+\infty) \colon 0\leq s\leq \rho(x)\}/\sim,\end{align*} $$

where the equivalence relation $\sim $ identifies $(x,\rho (x))$ with $(f(x),0)$ for all $x\in K$ . Denote $\pi $ the quotient map from $K\times [0,+\infty )$ to $K_{\rho }$ . We define the suspension flow over $f\colon K\to K$ with roof function $\rho $ by

$$ \begin{align*} f_t(x,s)=\pi(x,s+t). \end{align*} $$

For any function $g\colon K_{\rho }\to \mathbb {R}$ , we associate the function $\varphi _g\colon K\to \mathbb {R}$ by

$$ \begin{align*} \displaystyle\varphi_g(x)=\int_0^{\rho(x)}g(x,t)\,{d}t. \end{align*} $$

Since the roof function $\rho $ is continuous, $\varphi _g$ is continuous as long as g is. Moreover, to each invariant probability measure $\mu $ , we associate the measure $\mu _{\rho }$ given by

$$ \begin{align*} \displaystyle\int_{K_{\rho}}g\,{d}\mu_{\rho}=\frac{\int_K\varphi_g\, {d}\mu}{\int_K\rho\,{d}\mu} \quad \text{for all } g\in C(K_{\rho},\mathbb{R}). \end{align*} $$

Observe that not only the measure $\mu _{\rho }$ is $\mathfrak {F}$ -invariant (that is, $\mu _{\rho }(f_t^{-1}A)=\mu _{\rho }(A)$ for all $t\geq 0$ and measurable sets A), but also using that $\rho $ is bounded away from zero, the map

$$ \begin{align*} \mathcal{R}\colon\mathcal{M}(f,K)\to \mathcal{M}(\mathfrak{F},K_{\rho}) \quad\text{given by} \quad \mu\mapsto\mu_{\rho} \end{align*} $$

is a bijection. Abramov’s theorem [1, 31] states that $h_{\mu _{\rho }}(\mathfrak {F})=h_{\mu }(f)/\int \rho \,{d }\mu $ and hence, the topological entropy $h_{\mathrm {top}}(\mathfrak {F})$ of the flow satisfies

$$ \begin{align*} h_{\mathrm{top}}(\mathfrak{F})=\sup\{h_{\mu_{\rho}}(\mathfrak{F})\colon\mu_{\rho}\in \mathcal{M}(\mathfrak{F},K_{\rho})\}=\sup\bigg\{\frac{h_{\mu}(f)}{\int\rho\,{d}\mu}\colon\mu\in \mathcal{M}(f,K)\bigg\}. \end{align*} $$

Throughout, we will use the notation $\Phi =(\phi _t)_{t}$ for a flow on a compact metric space and $\mathfrak {F}=(f_t)_{t}$ for a suspension flow.

Consider a suspension flow $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ over a transitive subshift of finite type $(\Delta ,\sigma )$ with a continuous roof function $\rho .$ A metric on $\Delta $ is defined by $d(x, y)=2^{-m}$ if m is the largest positive integer with $x_{n}=y_{n}$ for any $|n|<m$ , and $d(x, y)=1$ if $x_{0} \neq y_{0}.$ There is a natural metric $d_{\Delta _{\rho }}$ , known as the Bowen–Walters metric and we have $d_{\Delta _{\rho }}((x,0),(y,0))=d(x,y)$ for any $x,y\in \Delta $ by [6]. Now we state a result on the ‘multi-horseshoe’ dense property of suspension flows.

Proposition 2.8. Suppose $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ is a suspension flow over a transitive subshift of finite type $(\Delta ,\sigma )$ with a continuous roof function $\rho .$ Then for any $F=\operatorname {\mathrm {cov}}\{\mu ^i_{\rho }\}_{i=1}^m\subseteq \mathcal {M}(\mathfrak {F}, \Delta _{\rho }),$ any $x\in \Delta $ , and any $\eta , \zeta ,\varepsilon>0$ , there exist compact $\mathfrak {F}$ -invariant subsets $\Delta ^i_{\rho }\subseteq \Xi _{\rho }\subsetneq \Delta _{\rho }$ such that for each $1\leq i\leq m$ :

1. (1) $f_{t}|_{\Delta ^i_{\rho }}: \Delta ^i_{\rho } \rightarrow \Delta ^i_{\rho }$ and $f_{t}|_{\Xi _{\rho }}: \Xi _{\rho }\rightarrow \Xi _{\rho }$ are suspension flows over transitive subshifts of finite type $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ with the roof function $\rho ;$

2. (2) $h_{\mathrm {top}}(\Delta ^i_{\rho })>h_{\mu _{\rho }^i}(\mathfrak {F})-\eta $ and consequently, $h_{\mathrm {top}}(\Xi _{\rho })>\sup \{h_{\kappa }(\mathfrak {F}):\kappa \in F\}-\eta ;$

3. (3) $d_H(F, \mathcal {M}(\mathfrak {F}, \Xi _{\rho }))<\zeta $ , $d_H(\mu _{\rho }^i, \mathcal {M}(\mathfrak {F}, \Delta ^i_{\rho }))<\zeta $ ;

4. (4) for any $z\in \Xi $ , there is an increasing sequence $\{t_m\}_{m=-\infty }^{+\infty }$ such that $f_{t_m}((z,0))\in \Delta \times \{0\}$ , $d_{\Delta _{\rho }}(f_{t_m}((z,0)),(x,0))<\varepsilon $ for any $m\in \mathbb {Z},$ and $\lim \nolimits _{m\to +\infty }t_m=\lim \nolimits _{m\to -\infty }-t_m=+\infty $ .

Proof. Each $\mathfrak {F}$ -invariant probability measure $\mu _{\rho }^i$ is determined by a $\sigma $ -invariant probability measure $\mu _i$ . Take $\tilde {\zeta },{\kern-1pt}\tilde {\eta }{\kern-1pt}>{\kern-1pt}0$ small enough such that if $\mu ,{\kern-1pt}\nu {\kern-1pt}\in{\kern-1pt} \mathcal {M}(\sigma ,{\kern-1pt}\Delta )$ satisfies $d^*(\mu ,{\kern-1pt}\nu ){\kern-1pt}<{\kern-1pt}\tilde {\zeta },$ then one has

(2.1) $$ \begin{align} d^*(\mathcal{R}(\mu),\mathcal{R}(\nu))<\zeta\quad \text{and}\quad\frac{\int \rho\, d\mu}{\int\rho \,d\nu}\bigg(h_{\mathcal{R}(\mu)}(\mathfrak{F})-\frac{2\tilde{\eta}}{\int\rho\,d \mu}\bigg)> h_{\mathcal{R}(\mu)}(\mathfrak{F})-\eta. \end{align} $$

For the $\tilde {F}=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\sigma ,\Delta )$ , $x\in \Delta $ and $\tilde {\eta }, \tilde {\zeta },\varepsilon>0,$ by Corollary 2.7, there exist compact invariant subsets $\Delta _i\subseteq \Xi \subsetneq \Delta $ such that for each $1\leq i\leq m$ :

1. (a) $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ are transitive subshifts of finite type;

2. (b) $h_{\mathrm {top}}(\sigma , \Delta _i)>h_{\mu _i}(\sigma )-\tilde {\eta }$ and consequently, $h_{\mathrm {top}}(\sigma , \Xi )>\sup \{h_{\kappa }(\sigma ):\kappa \in F\}-\tilde {\eta };$

3. (c) $d_H(\tilde {F}, \mathcal {M}(\sigma , \Xi ))<\tilde {\zeta }$ , $d_H(\mu _i, \mathcal {M}(\sigma , \Delta _i))<\tilde {\zeta }$ ;

4. (d) there is a positive integer L such that for any $z\in \Xi $ , one has $d(\sigma ^{j+mL}(z),x)<\varepsilon $ for some $0\leq j\leq L-1$ and any $m\in \mathbb {Z}$ .

Then by items (a) and (d), we have items (1) and (4). By equation (2.1) and item (c), we have $d_H(\mu _{\rho }^i, \mathcal {M}(\mathfrak {F}, \Delta ^i_{\rho }))<\zeta $ and thus F is contained in the $\zeta $ -neighborhood of $\mathcal {M}(\mathfrak {F}, \Xi _{\rho }).$ Note that for any $\theta _i\in [0, 1]$ with $\sum _{i=1}^m\theta _i=1$ , one has

$$ \begin{align*} \mathcal{R}\bigg(\sum_{i=1}^{m}\theta_i\mu_i\bigg)=\sum_{i=1}^{m}\frac{\theta_i\int \rho\,d\mu_i}{\sum_{j=1}^{m}\theta_j\int\rho\,d\mu_j}\mathcal{R}(\mu_i). \end{align*} $$

So $\mathcal {M}(\mathfrak {F}, \Xi _{\rho })$ is contained in the $\zeta $ -neighborhood of F and thus we have item (3).

By the variational principle of the topological entropy, there is $\nu _i\in \mathcal {M}(\sigma ,\Delta _i)$ such that $h_{\nu _i}(\sigma )>h_{\mathrm {top}}(\sigma , \Delta _i)-\tilde {\eta }>h_{\mu _i}(\sigma )-2\tilde {\eta }.$ Then

$$ \begin{align*} h_{\mathrm{top}}(\Delta^i_{\rho})\geq h_{\mathcal{R}(\nu_i)}(\mathfrak{F})=\frac{h_{\nu_i}(\sigma)}{\int \rho\,d\nu_i}>\frac{\int\rho\,{d}\mu_i}{\int\rho\,{d}\nu_i} \cdot \Big(h_{\mu^i_{\rho}}(\mathfrak{F}) -\frac{2\tilde{\eta}}{\int\rho\,{d}\mu_i}\Big) >h_{\mu^i_{\rho}}(\mathfrak{F}) -\eta \end{align*} $$

by equation (2.1). We obtain item (2).

2.3 Basic sets and proof of Theorem 2.5

Following the classical arguments of Bowen [7, 8] on Axiom A vector fields, every basic set is semi-conjugate to a suspension flow over a transitive subshift of finite type with a continuous roof function. Now we recall some basic results from [7, 8]. Readers can also see [6].

Given $X\in \mathscr {X}^1(M)$ and a basic set $\Lambda ,$ by [8, Theorem 2.5], there is a family of closed sets $R_{1}, \ldots , R_{k}$ and a positive number $\alpha $ so that the following properties hold:

1. (1) $\Lambda =\bigcup _{t \in [- \alpha ,0]} \phi _{t}(\bigcup _{i=1}^{k} R_{i});$

2. (2) there exists a transitive subshift of finite type $(\Delta ,\sigma )$ and a continuous and onto map $\pi :\Delta \to \bigcup _{i=1}^{k} R_{i}$ such that $\pi \circ \sigma =T\circ \pi $ (T is the transfer map whose definition will be recalled later);

3. (3) $\pi $ is one-to-one off $\bigcup _{n\in \mathbb {Z}} \sigma ^n(\pi ^{-1}(\bigcup _{i=1}^{k} \partial R_{i}))$ .

By item (1), we define the transfer function $\tau : \Lambda \rightarrow [0, \infty )$ by

$$ \begin{align*} \tau(x)=\min \bigg\{t>0: \phi_{t}(x) \in \bigcup_{i=1}^{k} R_{i}\bigg\}. \end{align*} $$

There is a positive number $\beta $ such that $\tau (x)\geq \beta $ for any $x\in \Lambda .$ Additionally, there is a metric on $\Delta $ such that $\tau \circ \pi $ is Lipschitz. Let $T: \Lambda \rightarrow \bigcup _{i=1}^{k} R_{i}$ be the transfer map given by $T(x)=\phi _{\tau (x)}(x)$ . The restriction of T to $\bigcup _{i=1}^{k} R_{i}$ is invertible and $\bigcup _{i=1}^{k} R_{i}$ is a T-invariant set. We can obtain a suspension flow $f_t:\Delta _{\rho }\to \Delta _{\rho }$ over the transitive subshift of finite type $(\Delta ,\sigma )$ with Lipschitz roof function $\rho =\tau \circ \pi .$ Then we extend $\pi $ to a finite-to-one surjection $\pi : \Delta _{\rho } \rightarrow \Lambda $ by $\pi (x, s)=(\phi _{s} \circ \pi )(x)$ for every $(x, s) \in \Delta _{\rho }.$ We have

$$ \begin{align*} \pi \circ f_{t}=\phi_{t} \circ \pi \end{align*} $$

and $\pi $ is one-to-one off $\bigcup _{t\in \mathbb {R}} f_t(\pi ^{-1}(\bigcup _{i=1}^{k} \partial R_{i}))$ .

The boundary of every $R_i$ consists of two parts $\partial R_i=\partial ^s R_i\cup \partial ^u R_i.$ Denote

$$ \begin{align*} \Delta^{s} \Lambda=\phi_{[0, \alpha]} \bigg(\bigcup_{i=1}^{k}\partial^s R_i\bigg)\quad\text{and}\quad \Delta^{u} \Lambda=\phi_{[-\alpha,0]} \bigg(\bigcup_{i=1}^{k}\partial^u R_i\bigg). \end{align*} $$

By [8, Proposition 2.6], one has $\phi _t(\Delta ^{s} \Lambda )\subset \Delta ^{s} \Lambda $ and $\phi _{-t}(\Delta ^{u} \Lambda )\subset \Delta ^{u} \Lambda $ for any $t\geq 0.$ In fact, in the proof of [8, Proposition 2.6], it is also shown that

(2.2) $$ \begin{align} T\bigg(\bigcup_{i=1}^{k}\partial^s R_i\bigg)\subset \bigcup_{i=1}^{k}\partial^s R_i\quad\text{and}\quad T^{-1}\bigg(\bigcup_{i=1}^{k}\partial^u R_i\bigg)\subset \bigcup_{i=1}^{k}\partial^u R_i. \end{align} $$

Now we give the proof of Theorem 2.5.

Proof of Theorem 2.5

Let $f_t:\Delta _{\rho }\to \Delta _{\rho }$ be the associated suspension flow of the basic set $\Lambda $ . Since $\pi ^{-1}(\bigcup _{i=1}^{k}\partial ^s R_i)$ is a proper closed subset of $\Delta ,$ then there is $\tilde {x}\in \Delta $ and $\tilde {\varepsilon }>0$ such that

$$ \begin{align*} B(\tilde{x},\tilde{\varepsilon})\cap\pi^{-1}\bigg( \bigcup_{i=1}^{k}\partial^s R_i\bigg)=\emptyset, \end{align*} $$

where $B(\tilde {x},\tilde {\varepsilon })=\{y\in \Delta : d(\tilde {x},y)<\tilde {\varepsilon }\}.$

Claim 2.9. If there is a point $z\in \Delta $ and an increasing sequence $\{t_m\}_{m=-\infty }^{+\infty }$ so that $f_{t_m}((z,0))\in \Delta \times \{0\}$ , $d_{\Delta _{\rho }}(f_{t_m}((z,0)),(\tilde {x},0))<\tilde {\varepsilon }$ for any $m\in \mathbb {Z},$ and $\lim \nolimits _{m\to +\infty }t_m=\lim \nolimits _{m\to -\infty }-t_m=+\infty ,$ then $z\notin \bigcup _{n\in \mathbb {Z}} \sigma ^n(\pi ^{-1}(\bigcup _{i=1}^{k} \partial R_{i})).$

Proof. Without loss of generality, we assume $\sigma ^l(z)\in \pi ^{-1}(\bigcup _{i=1}^{k} \partial ^s R_{i})$ for some integer $l\in \mathbb {Z}.$ Then by equation (2.2), we have $\sigma ^{l+n}(z)\in \pi ^{-1}(\bigcup _{i=1}^{k} \partial ^s R_{i})$ for any $n\geq 0.$ Note that $f_{t_m}((z,0))\in \Delta \times \{0\}$ implies there is an integer $n_m$ such that $f_{t_m}((z,0))=(\sigma ^{n_m}(z),0).$ So by $\lim \nolimits _{m\to +\infty }t_m=+\infty ,$ there is an integer $n_m>l$ such that $d(\sigma ^{n_m}(z),\tilde {x})=d_{\Delta _{\rho }}(f_{t_m}((z,0)),(\tilde {x},0))<\tilde {\varepsilon }.$ This contradicts $B(\tilde {x},\tilde {\varepsilon })\cap \pi ^{-1}( \bigcup _{i=1}^{k}\partial ^s R_i)=\emptyset .$

Fix $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\Phi ,\Lambda )$ and $\eta , \zeta>0.$ Then there is $\tilde {\mu }_i\in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ such that $\mu _i=\tilde {\mu }_i\circ \pi ^{-1}.$ Since $\pi $ is continuous, there is $\tilde {\zeta }>0$ such that $d^{*}(\tilde {\mu }\circ \pi ^{-1},\tilde {\nu }\circ \pi ^{-1})<\zeta $ for any $\tilde {\mu },\tilde {\nu }\in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ with $d^*(\tilde {\mu },\tilde {\nu })<\tilde {\zeta }.$ For $\tilde {F}=\operatorname {\mathrm {cov}}\{\tilde {\mu }_i\}_{i=1}^m\subseteq \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ , $\tilde {x}$ and $\eta ,\tilde {\zeta },\tilde {\varepsilon }>0,$ by Proposition 2.8, there exist compact $\mathfrak {F}$ -invariant subsets $\Delta ^i_{\rho }\subseteq \Xi _{\rho }\subsetneq \Delta _{\rho }$ such that for each $1\leq i\leq m$ :

1. (a) $f_{t}|_{\Delta ^i_{\rho }}: \Delta ^i_{\rho } \rightarrow \Delta ^i_{\rho }$ and $f_{t}|_{\Xi _{\rho }}: \Xi _{\rho }\rightarrow \Xi _{\rho }$ are suspension flows over transitive subshifts of finite type $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ with the roof function $\rho ;$

2. (b) $h_{\mathrm {top}}(\Delta ^i_{\rho })>h_{\tilde {\mu }_i}(\mathfrak {F})-\eta $ and consequently, $h_{\mathrm {top}}(\Xi _{\rho })>\sup \{h_{\kappa }(\mathfrak {F}):\kappa \in F\}-\eta ;$

3. (c) $d_H(\tilde {F}, \mathcal {M}(\mathfrak {F}, \Xi _{\rho }))<\tilde {\zeta }$ , $d_H(\tilde {\mu }_i, \mathcal {M}(\mathfrak {F}, \Delta ^i_{\rho }))<\tilde {\zeta }$ ;

4. (d) for any $z\in \Xi $ , there is an increasing sequence $\{t_m\}_{m=-\infty }^{+\infty }$ such that $f_{t_m}((z,0))\in \Delta \times \{0\}$ , $d_{\Delta _{\rho }}(f_{t_m}((z,0)),(\tilde {x},0))<\tilde {\varepsilon }$ for any $m\in \mathbb {Z},$ and $\lim \nolimits _{m\to +\infty }t_m=\lim \nolimits _{m\to -\infty }-t_m=+\infty $ .

By Claim 2.9 and item (d), we have

$$ \begin{align*}\Delta_i\cap \bigcup_{n\in\mathbb{Z}} \sigma^n\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset\quad\text{and}\quad\Xi\cap \bigcup_{n\in\mathbb{Z}} \sigma^n\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset. \end{align*} $$

Then

$$ \begin{align*}\Delta^i_{\rho}\cap \bigcup_{t\in\mathbb{R}} f_t\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset\quad\text{and}\quad \Xi_{\rho}\cap \bigcup_{t\in\mathbb{R}} f_t\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset. \end{align*} $$

This implies $\pi $ is one-to-one on $\Delta ^i_{\rho }$ and $\Xi _{\rho }.$ So $\Lambda _i=\pi (\Delta ^i_{\rho })$ and $\Theta =\pi (\Xi _{\rho })$ are the horseshoes we want. $\Box $

3 Asymptotically additive families and almost additive families

In this section, we consider Theorems A and B in the more general context of asymptotically additive families and almost additive families of continuous functions and give abstract conditions on which the results of Theorems A and B hold.

3.1 Non-additive thermodynamic formalism for flows

In this subsection, we recall a few basic notions and results on the non-additive thermodynamic formalism for flows. Consider a continuous flow $\Phi =(\phi _t)_{t\in \mathbb {R}}$ on a compact metric space $(M,d).$ Let $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Given $x\in \Lambda $ and $t,\varepsilon>0$ , we consider the set

$$ \begin{align*} B_{t}(x, \varepsilon)=\{y \in \Lambda: d(\phi_{s}(y), \phi_{s}(x))<\varepsilon \text { for any } s \in[0, t]\}. \end{align*} $$

Moreover, let $a=(a_t)_{t\geq 0}$ be a family of continuous functions $a_t:\Lambda \to \mathbb {R}$ with tempered variation, that is, such that

$$ \begin{align*} \lim _{\varepsilon \rightarrow 0} \limsup_{t \rightarrow \infty} \frac{\gamma_{t}(a, \varepsilon)}{t}=0, \end{align*} $$

where

$$ \begin{align*} \gamma_{t}(a, \varepsilon)=\sup \{|a_{t}(y)-a_{t}(x)|: y \in B_{t}(x, \varepsilon),\ x\in \Lambda \}. \end{align*} $$

Given $\varepsilon>0$ , we say that a set $\Gamma \subset \Lambda \times \mathbb {R}_{0}^{+}$ covers $\Lambda $ if

$$ \begin{align*} \bigcup_{(x, t) \in \Gamma} B_{t}(x, \varepsilon) \supset \Lambda, \end{align*} $$

and we write

$$ \begin{align*} a(x, t, \varepsilon)=\sup \{a_{t}(y): y \in B_{t}(x, \varepsilon)\} \quad \text { for }(x, t) \in \Gamma. \end{align*} $$

For each $\alpha \in \mathbb {R}$ , let

(3.1) $$ \begin{align} M(\Lambda, a, \alpha, \varepsilon)=\lim _{T \rightarrow+\infty} \inf _{\Gamma} \sum_{(x, t) \in \Gamma} \exp (a(x, t, \varepsilon)-\alpha t) \end{align} $$

with the infimum taken over all countable sets $\Gamma \subset \Lambda \times [T,+\infty )$ covering $\Lambda $ . When $\alpha $ goes from $-\infty $ to $+\infty $ , the quantity in equation (3.1) jumps from $+\infty $ to 0 at a unique value and so one can define

$$ \begin{align*} P(a, \Lambda, \varepsilon)=\inf \{\alpha \in \mathbb{R}: M(\Lambda, a, \alpha, \varepsilon)=0\}. \end{align*} $$

Moreover, the limit

$$ \begin{align*} P(a, \Lambda)=\lim _{\varepsilon \rightarrow 0} P(a, \Lambda, \varepsilon) \end{align*} $$

exists and is called the non-additive topological pressure of the family a on the set $\Lambda $ .

We recall that a family of functions $a=(a_t)_{t\geq 0}$ on $\Lambda $ is said to be almost additive with respect to the flow $(\phi _t)_{t\in \mathbb {R}}$ if there is a constant $C>0$ such that

$$ \begin{align*} -C+a_{t}+a_{s} \circ \phi_{t} \leqslant a_{t+s} \leqslant a_{t}+a_{s} \circ \phi_{t}+C \end{align*} $$

for every $t, s\geq 0$ . Let $A(\Phi ,\Lambda )$ be the set of all almost additive families of continuous functions $a=(a_{t})_{t \geqslant 0}$ on $\Lambda $ with tempered variation such that

$$ \begin{align*} \sup _{t \in[0, s]}\|a_{t}\|_{\infty}<\infty \quad \text {for some } s>0. \end{align*} $$

From [5, Proposition 4], the limit $\lim \nolimits _{t \rightarrow \infty } {1}/{t} \int _{\Lambda } a_{t}\,{d} \mu $ exists for any $a=(a_{t})_{t \geq 0}\in A(\Phi ,\Lambda )$ and any $\mu \in \mathcal {M}(\Phi ,\Lambda ),$ and the function

(3.2) $$ \begin{align} \mathcal{M}(\Phi,\Lambda) \ni \mu \mapsto \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu \end{align} $$

is continuous with the weak $^*$ topology. Let $a=(a_{t})_{t \geq 0}\in A(\Phi ,\Lambda ).$ Then by [4, Theorem 2.1 and Lemma 2.4], we have

(3.3) $$ \begin{align} \begin{aligned} P(a,\Lambda)&=\sup_{\mu\in\mathcal{M}(\Phi,\Lambda)}\bigg(h_{\mu}(\Phi)+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu\bigg)\\ &=\sup_{\mu\in\mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)}\bigg(h_{\mu}(\Phi)+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu\bigg). \end{aligned} \end{align} $$

A measure $\mu \in \mathcal {M}(\Phi ,\Lambda )$ is said to be an equilibrium measure for the almost additive family a if

$$ \begin{align*}P(a,\Lambda)=h_{\mu}(\Phi)+\lim\limits_{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu.\end{align*} $$

We say that a has bounded variation if for every $\kappa>0$ , there exists $\varepsilon>0$ such that

$$ \begin{align*} |a_{t}(x)-a_{t}(y)|<\kappa \quad \text { whenever } y \in B_{t}(x, \varepsilon). \end{align*} $$

Following the arguments of [4, §3.2], we have the following result on the uniqueness of equilibrium measure for suspension flows.

Lemma 3.1. Suppose $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ is a suspension flow over a transitive subshift of finite type $(\Delta ,\sigma )$ with a Lipschitz roof function $\rho .$ Let $a=(a_{t})_{t \geq 0}$ be an almost additive family of continuous functions on $\Delta _{\rho }$ with bounded variation such that $P(a,\Delta _{\rho })=0$ and $\sup _{t \in [0, s]}\|a_{t}\|_{\infty }<\infty $ for some $s>0$ . Then there is a unique equilibrium measure $\mu _{a}$ for a.

Proof. Define a sequence of functions $c_n:\Delta \to \mathbb {R}$ by

$$ \begin{align*}c_n(x)=a_{\rho_n(x)}(x),\end{align*} $$

where $\rho _n(x)=\sum _{i=0}^{n-1}\rho (\sigma ^i(x)).$ Following the arguments of [4, §3.2], we have:

1. (1) $c=(c_n)_{n\in \mathbb {N}}$ is an almost additive sequence of continuous functions, that is, there is a constant $\tilde {C}>0$ such that for every $n, m \in \mathbb {N}$ , we have

   $$ \begin{align*} -\tilde{C}+c_{n}+c_{m} \circ \sigma^{n} \leqslant c_{n+m} \leqslant \tilde{C}+c_{n}+c_{m} \circ \sigma^{n}; \end{align*} $$

2. (2) $c=(c_n)_{n\in \mathbb {N}}$ has bounded variation, that is, there exists $\varepsilon>0$ for which $ \sup _{n \in \mathbb {N}} \gamma _{n}(c, \varepsilon )<\infty $ with

   $$ \begin{align*}\gamma_{n}(c, \varepsilon)=\sup \{|c_{n}(x)-c_{n}(y)|: x,y\in\Delta,\ d(\sigma^{k} (x), \sigma^{k} (y))<\varepsilon \text { for } k=0, \ldots, n\};\end{align*} $$

3. (3) for any ergodic measure $\nu \in \mathcal {M}(\sigma ,\Delta )$ and $\mu =\mathcal {R}(\nu )$ (recall §2.2),

   $$ \begin{align*} h_{\mu}(\mathfrak{F})+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Delta_{\rho}} a_{t}\,d \mu=(h_{\nu}(\sigma)+\lim _{n \rightarrow \infty} \frac{1}{n} \int_{\Delta} c_{n}\,d \nu) \bigg/ \int_{\Delta} \rho\,d \nu. \end{align*} $$

By item (3), we have $h_{\mu }(\mathfrak {F})+\lim \nolimits _{t \rightarrow \infty } ({1}/{t}) \int _{\Delta _{\rho }} a_{t}\,d \mu =0$ if and only if $h_{\nu }(\sigma )+\lim \nolimits _{n \rightarrow \infty } ({1}/{n})\int _{\Delta } c_{n}\,d \nu =0.$ Since $P(a,\Delta _{\rho })=0$ , then $\mu $ is an equilibrium measure for a if and only if $\nu $ is an equilibrium measure for c by equation (3.3).

It is proved in [15, Theorem 5.3] that for every homeomorphism which is transitive, expansive, and has the shadowing property, and for every almost additive sequence of continuous functions with bounded variation, there is a unique equilibrium measure. Recall from [38] a subshift satisfies the shadowing property if and only if it is a subshift of finite type. As a subsystem of a two-side full shift, it is expansive. Then there is a unique equilibrium measure $\nu _{c}$ for c, and $\mathcal {R}(\mu _c)$ is the unique equilibrium measure for a.

Corollary 3.2. Suppose $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ is a suspension flow over a transitive subshift of finite type $(\Delta ,\sigma )$ with a Lipschitz roof function $\rho .$ Let $a=(a_{t})_{t \geq 0}$ be an almost additive family of continuous functions on $\Delta _{\rho }$ with bounded variation such that $\sup _{t \in [0, s]}\|a_{t}\|_{\infty }<\infty $ for some $s>0$ . Then there is a unique equilibrium measure $\mu _{a}$ for a.

Proof. For any $t \geq 0,$ define

$$ \begin{align*} b_{t}=a_{t}-P(a) t. \end{align*} $$

Then b is an almost additive family with bounded variation and satisfies $\sup _{t \in [0, s]}\|b_{t}\|_{\infty }<\infty $ and $P(b)=0$ . For each $\mathfrak {F}$ -invariant probability measure $\mu $ on $\Delta _{\rho }$ , we have

$$ \begin{align*} \frac{1}{t} \int_{X} a_{t}\,d \mu=\frac{1}{t} \int_{X} b_{t}\,d \mu-P(a). \end{align*} $$

This implies that a and b have the same equilibrium measures. Then by Lemma 3.1, we complete the proof.

Lemma 3.3. Given $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , assume that there exists a suspension flow $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ over a transitive subshift of finite type $(\Delta ,\sigma )$ with a Lipschitz roof function $\rho $ , and a finite-to-one continuous surjection $\pi : \Delta _{\rho } \rightarrow \Lambda $ , such that $\pi \circ f_t=\phi _t\circ \pi $ . Then for every almost additive family of continuous functions $a=(a_{t})_{t \geq 0}$ on $\Lambda $ with bounded variation such that $\sup _{t \in [0, s]}\|a_{t}\|_{\infty }<\infty $ for some $s>0,$ there is a unique equilibrium measure $\mu _{a}$ for a.

Proof. Define $\tilde {a}_t=a_t\circ \pi $ for any $t\geq 0.$ Since $\pi $ is continuous, $\tilde {a}=(\tilde {a}_{t})_{t \geq 0}$ is an almost additive family of continuous functions on $\Delta _{\rho }$ with bounded variation and $\sup _{t \in [0, s]}\|\tilde {a}_{t}\|_{\infty }<\infty .$ By Corollary 3.2, there is a unique equilibrium measure $\mu _{\tilde {a}}$ for $\tilde {a}$ . Since $\pi $ is finite-to-one, we have $h_{\mu _{\tilde {a}}\circ \pi ^{-1}}(\Phi )=h_{\mu _{\tilde {a}}}(\mathfrak {F})$ and thus

$$ \begin{align*}h_{\mu_{\tilde{a}}\circ\pi^{-1}}(\Phi)+\lim\limits_{t\to\infty}\frac{1}{t}\int a_t\, d\mu_{\tilde{a}}\circ\pi^{-1}=h_{\mu_{\tilde{a}}}(\mathfrak{F})+\lim\limits_{t\to\infty}\frac{1}{t}\int \tilde{a}_t\, d\mu_{\tilde{a}}=P(\tilde{a})\geq P(a).\end{align*} $$

Then by equation (3.3), $\mu _{\tilde {a}}\circ \pi ^{-1}$ is an equilibrium measure for $a.$ To show that the measure is unique, suppose $\nu $ is an equilibrium measure for $a,$ and choose $\mu \in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ with $\mu \circ \pi ^{-1}=\nu .$ Then

$$ \begin{align*}h_{\mu}(\mathfrak{F})+\lim\limits_{t\to\infty}\frac{1}{t}\int \tilde{a}_t\,d\mu=h_{\nu}(\Phi)+\lim\limits_{t\to\infty}\frac{1}{t}\int a_t\,d\nu= P(a)=P(\tilde{a}).\end{align*} $$

Thus, $\mu =\mu _{\tilde {a}}$ by the uniqueness of $\mu _{\tilde {a}}.$ So $\nu =\mu \circ \pi ^{-1}=\mu _{\tilde {a}}\circ \pi ^{-1}$ is the unique equilibrium measure for a.

Then we have the following theorem, combining §2.3 and Definition 2.2.

Theorem 3.4. Given $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , if $\Lambda $ is a basic set or a horseshoe of $\phi _{t},$ then for every almost additive family of continuous functions $a=(a_{t})_{t \geq 0}$ on $\Lambda $ with bounded variation such that $\sup _{t \in [0, s]}\|a_{t}\|_{\infty }<\infty $ for some $s>0,$ there is a unique equilibrium measure $\mu _{a}$ for a.

3.2 Conditional variational principles

3.2.1 Conditional variational principle of almost additive families

Consider a continuous flow $\Phi =(\phi _t)_{t\in \mathbb {R}}$ on a compact metric space $(M,d).$ Let $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ , where

$$ \begin{align*} A=(a^{1}, \ldots, a^{d}) \quad\text {and}\quad B=(b^{1}, \ldots, b^{d}), \end{align*} $$

and $a^{i}=(a_{t}^{i})_{t\in \mathbb {R}}$ and $b^{i}=(b_{t}^{i})_{t\in \mathbb {R}}$ . Assume that

(3.4) $$ \begin{align} \liminf _{t \rightarrow \infty} \frac{b_{t}^{i}(x)}{t}>0 \quad \text {and} \quad b_{t}^{i}(x)>0 \end{align} $$

for every $i=1, \ldots , d, x \in \Lambda $ , and $t \in \mathbb {R}$ . Given $\alpha =(\alpha _1, \ldots , \alpha _{d}) \in \mathbb {R}^{d}$ , we define

$$ \begin{align*} R_{A,B}(\alpha)=\bigcap_{i=1}^{d}\bigg\{x \in \Lambda: \lim _{t \rightarrow \infty} \frac{a_{t}^{i}(x)}{b_{t}^{i}(x)}=\alpha_{i}\bigg\}. \end{align*} $$

We also define the map $\mathcal {P}_{A,B}: \mathcal {M}(\Phi ,\Lambda ) \rightarrow \mathbb {R}$ by

(3.5) $$ \begin{align} \mathcal{P}_{A,B}(\mu) =\lim _{t \rightarrow \infty}\bigg(\frac{\int a_{t}^{1}\,d \mu}{\int b_{t}^{1}\,d \mu}, \ldots, \frac{\int a_{t}^{d}\,d \mu}{\int b_{t}^{d}\,d \mu}\bigg). \end{align} $$

Equation (3.2) ensures that the function $\mathcal {P}_{A,B}$ is continuous. Denote

$$ \begin{align*}L_{A,B}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{M}(\Phi,\Lambda)\}.\end{align*} $$

Let $E(\Phi ,\Lambda ) \subset A(\Phi ,\Lambda )$ be the set of families with a unique equilibrium measure. Define the sequence of constant functions $u=(u_t)_{t\geq 0}$ with $u_t\equiv t$ for any $t\geq 0.$ In [5], L. Barreira and C. Holanda give the conditional variational principle as the following theorem.

Theorem 3.5. [5, Theorem 9]

Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d)$ and $\Lambda \subset M$ is a compact invariant set such that the entropy function $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4) and $ \text {span}\{a^1, b^1,\ldots ,a^d,b^d,u\} \subseteq E(\Phi ,\Lambda ). $ If $\alpha \in \mathrm {Int}(L_{A,B})$ , then $R_{A,B}(\alpha )\neq \emptyset $ , and the following properties hold.

1. (1) $h_{\mathrm {top}}(R_{A,B}(\alpha ))$ satisfies the variational principle:

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=\max \{h_{\mu}(\Phi): \mu \in \mathcal{M}(\Phi,\Lambda) \text { and } \mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

2. (2) There is an ergodic measure $\mu _{\alpha } {\kern-1pt}\in{\kern-1pt} \mathcal {M}(\Phi ,\Lambda ) $ with $\mathcal {P}_{A,B}(\mu _{\alpha })=\alpha , \mu _{\alpha }(R_{A,B}(\alpha ))=1$ , and

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=h_{\mu_{\alpha}}(\Phi). \end{align*} $$

Remark 3.6. In fact, Barreira and Holanda in [5, Theorem 9] give the proof of Theorem 3.5 under the assumption $d=1.$ However, Barreira and Doutor in [3, Theorem 3] proved the case of discrete time for any $d\geq 1.$ Combining the two proofs, one can obtain Theorem 3.5 for any $d\geq 1.$

3.2.2 Conditional variational principle of asymptotically additive families

Let $\Phi =(\phi _t)_{t \in \mathbb {R}}$ be a continuous flow on a compact metric space $(M,d)$ and $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. A family of functions $a=(a_t)_{t \geq 0}$ is said to be asymptotically additive with respect to $\Phi $ on $\Lambda $ if for each $\varepsilon>0$ , there exists a function $b_{\varepsilon }: \Lambda \rightarrow \mathbb {R}$ such that

$$ \begin{align*} \limsup _{t \rightarrow \infty} \frac{1}{t}\sup_{x\in\Lambda}\bigg|a_t(x)-\int_0^t(b_{\varepsilon} \circ \phi_s(x))\,d s\bigg| \leq \varepsilon. \end{align*} $$

Let $AA(\Phi ,\Lambda )$ be the set of all asymptotically additive families of continuous functions $a=(a_{t})_{t \geqslant 0}$ on $\Lambda $ with tempered variation such that

$$ \begin{align*} \sup _{t \in[0, s]}\|a_{t}\|_{\infty}<\infty \quad \text {for some } s>0. \end{align*} $$

Proceeding as in [16], one can see that every almost additive family of functions is asymptotically additive.

Now assume that $\Phi $ is expansive on $\Lambda $ . Holanda in [20, Corollary 6] proved that for any $a=(a_t)_{t \geq 0}\in AA(\Phi ,\Lambda ),$ there exists a continuous function $b: \Lambda \rightarrow \mathbb {R}$ such that

$$ \begin{align*} \lim \limits_{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu=\int_{\Lambda} b\,d\mu \end{align*} $$

for any $\mu \in \mathcal {M}(\Phi ,\Lambda ).$ This implies that the function

(3.6) $$ \begin{align} \mathcal{M}(\Phi,\Lambda) \ni \mu \mapsto \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu \end{align} $$

is continuous with the weak $^*$ topology. Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , where

$$ \begin{align*} A=(a^{1}, \ldots, a^{d}) \quad\text {and}\quad B=(b^{1}, \ldots, b^{d}), \end{align*} $$

and $a^{i}=(a_{t}^{i})_{t\in \mathbb {R}}$ and $b^{i}=(b_{t}^{i})_{t\in \mathbb {R}}$ . Assume that

(3.7) $$ \begin{align} \begin{aligned} &\lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} b_t^i\,d \mu \geq 0 \quad \text {for all } \mu \in \mathcal{M}(\Phi,\Lambda) \end{aligned} \end{align} $$

with equality only permitted when

(3.8) $$ \begin{align} \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_t^i\,d \mu \neq 0 \end{align} $$

for every $i=1, \ldots , d.$ Given $\alpha =(\alpha _1, \ldots , \alpha _{d}) \in \mathbb {R}^{d}$ , we define

$$ \begin{align*} R_{A,B}(\alpha)=\bigcap_{i=1}^{d}\bigg\{x \in \Lambda: \lim _{t \rightarrow \infty} \frac{a_{t}^{i}(x)}{b_{t}^{i}(x)}=\alpha_{i}\bigg\}. \end{align*} $$

We also define the map $\mathcal {P}_{A,B}: \mathcal {M}(\Phi ,\Lambda ) \rightarrow \mathbb {R}$ by

(3.9) $$ \begin{align} \mathcal{P}_{A,B}(\mu) =\lim _{t \rightarrow \infty}\bigg(\frac{\int a_{t}^{1}\,d \mu}{\int b_{t}^{1}\,d \mu}, \ldots, \frac{\int a_{t}^{d}\,d \mu}{\int b_{t}^{d}\,d \mu}\bigg). \end{align} $$

Equation (3.6) ensures that the function $\mathcal {P}_{A,B}$ is continuous. Denote

$$ \begin{align*}L_{A,B}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{M}(\Phi,\Lambda)\}.\end{align*} $$

Without using the uniqueness of equilibrium assumption, C. Holanda obtains a conditional variational principle for asymptotically additive families of continuous functions.

Theorem 3.7. [20, Theorem 11]

Given $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , assume that $\Lambda $ is a basic set or a horseshoe of $\phi _{t}.$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). If $\alpha \in \mathrm {Int}(L_{A,B})$ , then $R_{A,B}(\alpha )\neq \emptyset $ , and the following properties hold.

1. (1) $h_{\mathrm {top}}(R_{A,B}(\alpha ))$ satisfies the variational principle:

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=\sup \{h_{\mu}(\Phi): \mu \in \mathcal{M}(\Phi,\Lambda) \text { and } \mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

2. (2) For any $\varepsilon>0$ , there exists an ergodic measure $\mu _{\alpha } \in \mathcal {M}(\Phi ,\Lambda ) $ such that $\mathcal {P}_{A,B}(\mu _{\alpha })=\alpha , \mu _{\alpha }(R_{A,B}(\alpha ))=1$ and

   $$ \begin{align*} |h_{\mathrm{top}}(R_{A,B}(\alpha))-h_{\mu_{\alpha}}(\Phi)|<\varepsilon. \end{align*} $$

Remark 3.8. Using the work of Climenhaga [11, Theorem 3.3] and Cuneo [13, Theorem 1.2], and the conclusion that every Hölder continuous function has a unique equilibrium measure with respect to the map T (recall that T is the transfer map, see §2.3), Holanda in [20, Theorem 11] gives the proof of Theorem 3.7 under the assumption $d=1$ and $\Lambda $ is a mixing basic set. However, [11, Theorem 3.3] is stated for any $d\geq 1$ and by [9, Example 2], every Hölder continuous function has a unique equilibrium measure with respect to the map T when $\Lambda $ is just a transitive basic set. So one can obtain Theorem 3.7 for any $d\geq 1$ and any basic set $\Lambda .$ When $\Lambda $ is a horseshoe of $\phi _{t},$ we can verify the results also hold following the argument of [20, Theorem 11].

3.3 Abstract conditions on which Theorems A and B hold

Let $\Phi =(\phi _t)_{t \in \mathbb {R}}$ be a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Assume that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}.$ For any $\alpha \in L_{A,B},$ denote

$$ \begin{align*}M_{A,B}(\alpha)&=\{\mu\in \mathcal{M}(\Phi,\Lambda):\mathcal{P}_{A,B}(\mu)=\alpha\}, \\M_{A,B}^{\mathrm{erg}}(\alpha) &=\{\mu\in \mathcal{M}_{\mathrm{erg}} (\Phi,\Lambda):\mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

Then $M_{A,B}(\alpha )$ is closed in $\mathcal {M}(\Phi ,\Lambda )$ since the map $\mathcal {P}_{A,B}$ is continuous.

Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. We define the pressure of $\chi $ with respect to $\mu $ by $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\chi (\mu ).$ Now we give a result in the context of asymptotically additive families.

Theorem 3.9. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. Assume that the following holds: for any $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\Phi ,\Lambda )$ and any $\eta , \zeta>0$ , there are compact invariant subsets $\Lambda _i\subseteq \Theta \subsetneq \Lambda $ such that for each $i\in \{1,2,\ldots ,m\}$ :

1. (1) for any $a\in \mathrm {Int}(\mathcal {P}_{A,B}(\mathcal {M}(\Phi ,\Theta )))$ and any $\varepsilon>0,$ there exists an ergodic measure $\mu _a$ supported on $\Theta $ with $\mathcal {P}_{A,B}(\mu _a)=a$ such that $|h_{\mu _a}(\Phi )-H(\Phi ,a,\Theta )|<\varepsilon ,$ where $H(\Phi ,a,\Theta )=\sup \{h_{\mu }(\Phi ):\mu \in \mathcal {M}(\Phi ,\Theta ) \text { and }\mathcal {P}_{A,B}(\mu )=a\};$

2. (2) $h_{\mathrm {top}}(\Lambda _i)>h_{\mu _i}(\Phi )-\eta ;$

3. (3) $d_H(K, \mathcal {M}(\Phi , \Theta ))<\zeta $ , $d_H(\mu _i, \mathcal {M}(\Phi , \Lambda _i))<\zeta .$

Then for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu _0\in M_{A,B}(\alpha )$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$

For any $\alpha \in L_{A,B},$ denote

$$ \begin{align*}H_{A,B}(\Phi,\chi,\alpha)=\sup\{P(\Phi,\chi,\mu):\mu\in M_{A,B}(\alpha)\}.\end{align*} $$

In particular, when $\chi \equiv 0,$ we write

$$ \begin{align*}H_{A,B}(\Phi,\alpha)=H_{A,B}(\Phi,0,\alpha)=\sup\{h_{\mu}(\Phi):\mu\in M_{A,B}(\alpha)\}.\end{align*} $$

We list two conditions for $\chi :$

1. (A.1) for any $\mu _1, \mu _2 \in \mathcal {M}(\Phi ,\Lambda )$ with $P(\Phi ,\chi ,\mu _1) \neq P(\Phi ,\chi ,\mu _2)$ ,

   (3.10) $$ \begin{align} \beta(\theta):=P(\Phi,\chi,\theta \mu_1+(1-\theta) \mu_2)\text{ is strictly monotonic on }[0,1]; \end{align} $$

2. (A.2) for any $\mu _1, \mu _2 \in \mathcal {M}(\Phi ,\Lambda )$ with $P(\Phi ,\chi ,\mu _1) = P(\Phi ,\chi ,\mu _2)$ ,

   (3.11) $$ \begin{align} \beta(\theta):=P(\Phi,\chi,\theta \mu_1+(1-\theta) \mu_2)\text{ is constant on }[0,1]. \end{align} $$

Now we give abstract conditions on which Theorems A and B hold in the context of asymptotically additive families.

Theorem 3.10. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function satisfying equations (3.10) and (3.11). Assume that for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu _0\in M_{A,B}(\alpha )$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$ If $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,\Lambda ),$ then we have the following.

1. (1) For any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu _0\in M_{A,B}(\alpha ),$ any $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c\leq P(\phi ,\chi ,\mu _0)$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-c|<\eta .$

2. (2) For any $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )=c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ If further there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ , then for any $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )=c,\ S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$

3. (3) The set $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu \in \mathcal {M}(\Phi ,\Lambda ),\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )}\chi (\mu ) \leq P(\Phi ,\chi ,\mu )< H_{A,B}(\Phi ,\chi ,\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu \in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ),\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}(\Phi ,\chi ,\alpha )\}.$

Example 3.11. The function $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ can be defined as follows.

1. (1) $\chi \equiv 0.$ Then $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )$ is the metric entropy of $\mu .$

2. (2) $\chi (\mu )=\int g\,d \mu $ with a continuous function $g.$ Then from the weak $^*$ -topology on $\mathcal {M}(\Phi ,\Lambda )$ , $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ is a continuous function. Here, $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\chi (\mu )$ is the pressure of g with respect to $\mu .$

3. (3) $\chi (\mu )=\lim \nolimits _{t \rightarrow \infty } ({1}/{t}) \int a_{t}\,d \mu $ with an asymptotically additive family of continuous functions $a=(a_{t})_{t\in \mathbb {R}}$ on $\Lambda .$ Then $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ is a continuous function from equation (3.6). Here, $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\lim \nolimits _{t \rightarrow \infty } ({1}/{t}) \int a_{t}\,d \mu $ is the pressure of a with respect to $\mu .$

Furthermore, if $\chi $ is defined as above, then equations (3.10) and (3.11) hold for $\chi $ since it is affine.

Remark 3.12. In Theorems 3.9 and 3.10, the expansivity of $\Phi $ is used to guarantee the function $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto \lim _{t \rightarrow \infty } ({1}/{t}) \int _{\Lambda } a_{t}\,{d} \mu $ is continuous and the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. When we consider almost additive families, by equation (3.2), we only need to assume that the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous.

3.4 Proof of Theorem 3.9

3.4.1 Some lemmas

We establish several auxiliary results. For any $r\in \mathbb {R},$ denote $r^+=\{s\in \mathbb {R}:s>r\}$ and $r^-=\{s\in \mathbb {R}:s<r\}.$ For any $d \in \mathbb {N}$ , $r=(r_1, \ldots , r_{d})\in \mathbb {R}^d$ , and $\xi =(\xi _1, \ldots , \xi _{d})\in \{+,-\}^d$ , we define

$$ \begin{align*}r^{\xi}=\{s=(s_1, \ldots, s_{d})\in\mathbb{R}^d:s_i\in r_i^{\xi_i} \text{ for }i=1,2,\ldots,d\}.\end{align*} $$

We denote $F^d=\{({p_1}/{q_1}, \ldots , {p_d}/{q_d}):p_i,q_i\in \mathbb {R}\text { and }q_i>0 \text { for any }1\leq i\leq d\}.$ It is easy to check the following lemma.

Lemma 3.13. Let $b_i={p^i}/{q^i}\in F^1$ for $i=1,2.$

1. (1) If $b_1=b_2,$ then $({\theta p^1+(1-\theta )p^2})/({\theta q^1+(1-\theta )q^2})=b_1=b_2$ for any $\theta \in [0,1].$

2. (2) If $b_1\neq b_2,$ then $({\theta p^1+(1-\theta )p^2})/({\theta q^1+(1-\theta )q^2})$ is strictly monotonic on $\theta \in [0,1].$

We can obtain the following result using mathematical induction.

Lemma 3.14. Let $d \in \mathbb {N}$ and $a=({p_1}/{q_1}, \ldots , {p_d}/{q_d}) \in F^{d}.$ If $\{b_{\xi }=({p_1^{\xi }}/{q_1^{\xi }}, \ldots , {p_d^{\xi }}/ {q_d^{\xi }})\}_{\xi \in \{+,-\}^d}\subseteq F^d$ are $2^d$ numbers satisfying $b_{\xi }\in a^{\xi }$ for any $\xi \in \{+,-\}^d,$ then there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and

$$ \begin{align*} \frac{\sum_{\xi\in\{+,-\}^d}\theta_{\xi} p^{\xi}_i}{\sum_{\xi\in\{+,-\}^d}\theta_{\xi} q^{\xi}_i}=\frac{p_i}{q_i}\quad\text{for any }1\leq i\leq d.\end{align*} $$

From Lemma 3.14, we have the following corollary.

Corollary 3.15. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , satisfying equation (3.7). Then for any $\alpha \in \mathrm {Int}(L_{A,B}),$ and $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ with

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{for any }\xi\in \{+,-\}^d,\end{align*} $$

there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that

$$ \begin{align*} \sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1 \quad \text{and}\quad \mathcal {P}_{A,B} \left(\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\mu _{\xi }\right) = \alpha . \end{align*} $$

Lemma 3.16. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Then for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu \in M_{A,B}(\alpha )$ , and any $\eta ,\zeta>0$ , there are $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that for any $\xi \in \{+,-\}^d$ ,

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi}, h_{\mu_{\xi}}(\Phi)>h_{\mu}(f)-\eta \quad\text{and}\quad d^*(\mu_{\xi},\mu)<\zeta.\end{align*} $$

Proof. By $\alpha \in \mathrm {Int}(L_{A,B})$ , there is $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ such that $\mathcal {P}_{A,B}(\nu _{\xi })\in a^{\xi }$ for any $\xi \in \{+,-\}^d.$ We choose $\tau _{\xi }\in (0,1)$ close to $1$ such that $\mu _{\xi }=\tau _{\xi }\mu +(1-\tau _{\xi })\nu _{\xi }$ satisfies

$$ \begin{align*}h_{\mu_{\xi}}(\Phi)>h_{\mu}(\Phi)-\eta \quad\text{and}\quad d^*(\mu_{\xi},\mu)<\zeta \quad\text{for any }\xi\in \{+,-\}^d.\end{align*} $$

Then we have $\mathcal {P}_{A,B}(\mu _{\xi })\in \alpha ^{\xi }$ by $\tau ^{\xi }>0$ and Lemma 3.13(2).

3.4.2 Proof of Theorem 3.9

Fix $\alpha \in \mathrm {Int}(L_{A,B})$ , $\mu _0\in M_{A,B}(\alpha )$ , and $\eta , \zeta>0.$ Since $\Phi $ is expansive on $\Lambda ,$ the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Hence, there is $0<\zeta '<\zeta $ such that for any $\omega \in \mathcal {M}(\Phi ,\Lambda )$ with $d^*(\mu _0,\omega )<\zeta '$ , we have

(3.12) $$ \begin{align} h_{\omega}(\Phi)<h_{\mu_0}(\Phi)+\frac{3\eta}{8} \quad\text{and}\quad|\chi(\omega)-\chi(\mu_0)|<\frac{\eta}{2}. \end{align} $$

By Lemma 3.16, there are $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that for any $\xi \in \{+,-\}^d$ ,

(3.13) $$ \begin{align} \mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi},\quad h_{\mu_{\xi}}(\Phi)>h_{\mu_0}(\Phi)-\frac{\eta}{8} \quad\text{and}\quad d^*(\mu_{\xi},\mu_0)<\frac{\zeta'}{2}. \end{align} $$

Since the map $\mathcal {P}_{A,B}$ is continuous, there is $0<\zeta "<\zeta '$ such that for any $\omega _{\xi } \in \mathcal {M}(\Phi ,\Lambda )$ with $d^*(\omega _{\xi },\mu _{\xi })<\zeta "$ , one has

(3.14) $$ \begin{align} \mathcal{P}_{A,B}(\omega_{\xi})\in \alpha^{\xi}. \end{align} $$

For the $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d},$ there are compact invariant subsets $\Lambda _{\xi }\subseteq \Theta \subsetneq \Lambda $ such that for each $\xi \in \{+,-\}^d$ :

1. (1) for any $a\in \mathrm {Int}(\mathcal {P}_{A,B}(\mathcal {M}(\Phi ,\Theta )))$ and any $\varepsilon>0,$ there exists an ergodic measure $\mu _a$ supported on $\Theta $ with $\mathcal {P}_{A,B}(\mu _a)=a$ such that $|h_{\mu _a}(\Phi )-H(\Phi ,a,\Theta )|<\varepsilon ;$

2. (2) $h_{\mathrm {top}}( \Lambda _{\xi })>h_{\mu _{\xi }}(\Phi )-{\eta }/{8};$

3. (3) $d_H(\operatorname {\mathrm {cov}}\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}, \mathcal {M}(\Phi ,\Theta ))<{\zeta "}/{2}$ , $d_H(\mu _{\xi }, \mathcal {M}(\Phi ,\Lambda _{\xi }))<{\zeta "}/{2}.$

By item (2) and the variational principle, there is $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda _{\xi })$ such that

$$ \begin{align*}h_{\nu_{\xi}}(\Phi)>h_{\mathrm{top}}( \Lambda_{\xi})-\frac{\eta}{8}>h_{\mu_{\xi}}(\Phi)-\frac{2\eta}{8}>h_{\mu_0}(\Phi)-\frac{3\eta}{8}.\end{align*} $$

Then by item (3) and equation (3.14), we have $\mathcal {P}_{A,B}(\nu _{\xi })\in \alpha ^{\xi }.$ By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and $\mathcal {P}_{A,B}(\nu ')= \alpha $ , where $\nu '=\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\nu _{\xi }.$ Then on one hand, we have

(3.15) $$ \begin{align} H(\Phi,\alpha,\Theta) \geq h_{\nu'}(\Phi)\geq \min\{h_{\nu_{\xi}}(\Phi):\xi\in \{+,-\}^d\}> h_{\mu_0}(\Phi)-\frac{3\eta}{8}. \end{align} $$

On the other hand, by item (3), and equations (3.13) and (3.12), we have

(3.16) $$ \begin{align} H(\Phi,\alpha,\Theta)<h_{\mu_0}(\Phi)+\frac{3\eta}{8}. \end{align} $$

Now by item (1), there exists an ergodic measure $\nu $ supported on $\Theta $ with $\mathcal {P}_{A,B}(\nu )=\alpha $ such that $|h_{\nu }(\Phi )-H(\Phi ,\alpha ,\Theta )|<{\eta }/{8}.$ Then $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ , and by equations (3.15) and (3.16), we have $|h_{\nu }(\Phi )-h_{\mu _0}(\Phi )|<{\eta }/{2}.$ By item (3) and equation (3.13), we have $d^*(\nu ,\mu _0)<\zeta '<\zeta .$ Finally, by equation (3.12), we have $|\chi (\nu )-\chi (\mu _0)|<{\eta }/{2}$ and thus $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$

3.5 Proof of Theorem 3.10

3.5.1 Some lemmas

Lemma 3.17. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set. Let V be a convex subset of $\mathcal {M}(\Phi ,\Lambda ).$ If there is an invariant measure $\mu _V\in V$ with $S_{\mu _V}=\Lambda ,$ then $\{\mu \in V:S_{\mu }=\Lambda \}$ is residual in $V.$

Proof. Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is either empty or a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda )$ from [14, Proposition 21.11], if there is an invariant measure $\mu _V\in V$ with $S_{\mu _V}=\Lambda ,$ then $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,M).$ Thus, $\{\mu \in V:S_{\mu }=\Lambda \}$ is a $G_{\delta }$ subset of $V.$ In addition, for any $\nu \in V$ and $\theta \in (0,1),$ we have $\nu _{\theta }=\theta \nu + (1-\theta )\mu _V\in V$ and $S_{\nu _{\theta }}=\Lambda .$ So $\{\mu \in V:S_{\mu }=\Lambda \}$ is dense in $V,$ and thus is residual in $V.$

Lemma 3.18. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , satisfying equation (3.7). If $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ is a continuous function satisfying equations (3.10) and (3.11), then for any $\alpha \in \mathrm {Int}(L_{A,B})$ and any $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the following properties hold:

1. (1) if $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\},$ then $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ){\kern-1pt}\geq{\kern-1pt} c\}$ is residual in $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt} \mu ){\kern-1pt}\geq{\kern-1pt} c\};$

2. (2) if there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}{\kern-1pt}={\kern-1pt}\Lambda $ , then $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ){\kern-1pt}\geq{\kern-1pt} c, S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\};$

3. (3) if $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,\Lambda ),$ then $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )= c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$

Proof. (1) From [14, Proposition 5.7], $\mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda )$ is a $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda ).$ Then $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is a $G_{\delta }$ subset of $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ If $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\},$ then $\{\mu \in M_{A,B}^{\mathrm {erg}}y(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$

(2) Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is either empty or a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda )$ from [14, Proposition 21.11], if there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ , then $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda ).$

Now we show that $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c,\ S_{\mu }=\Lambda \}$ is non-empty. By Lemma 3.16, there exist $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{for any }\xi\in \{+,-\}^d.\end{align*} $$

Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is dense in $ \mathcal {M}(\Phi ,\Lambda )$ and the map $\mathcal {P}_{A,B}$ is continuous, then there exists $\omega _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ close to $\mu _{\xi }$ such that

$$ \begin{align*} \mathcal{P}_{A,B}(\omega_{\xi})\in \alpha^{\xi},\quad S_{\omega_{\xi}}=\Lambda\quad\text{for any }\xi\in \{+,-\}^d. \end{align*} $$

By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and $\mathcal {P}_{A,B}(\omega )= \alpha $ , where $\omega =\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\mu _{\omega }$ . Then $S_{\omega }=\Lambda .$ Since $c< H_{A,B}(\Phi ,\chi ,\alpha ),$ there is $\nu \in M_{A,B}(\alpha )$ such that $P(\Phi ,\chi ,\nu )>c.$ By equation (3.10), we can choose $\theta \in (0,1)$ close to $1$ such that $\mu '=\theta \nu +(1-\theta )\omega $ satisfies $P(\Phi ,\chi ,\mu ')>c.$ Then $\mu '\in \{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c,\ S_{\mu }=\Lambda \}.$ Note that $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is a convex set by equations (3.10), (3.11), and Lemma 3.13. So by Lemma 3.17, we complete the proof of item (2).

(3) Fix $\mu _0\in M_{A,B}(\alpha )$ with $P(\Phi ,\chi ,\mu _0)\geq c$ and $\zeta>0.$ By Lemma 3.16, there exist $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that

(3.17) $$ \begin{align} \mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{and}\quad d^*(\mu_{\xi},\mu_0)<\frac{\zeta}{2} \quad\text{for any }\xi\in \{+,-\}^d. \end{align} $$

Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,\Lambda )$ and the function $\mathcal {P}_{A,B}$ is continuous, then there exists $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ close to $\mu _{\xi }$ such that

(3.18) $$ \begin{align} \mathcal{P}_{A,B}(\nu_{\xi})\in \alpha^{\xi},\quad h_{\nu_{\xi}}(\Phi)=0 \quad\text{and}\quad d^*(\nu_{\xi},\mu_{\xi})<\frac{\zeta}{2} \quad\text{for each } \xi\in \{+,-\}^d. \end{align} $$

By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d} \theta _{\xi } =1$ and $\mathcal {P}_{A,B}(\nu ')= \alpha $ , where $\nu '=\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\nu _{\xi }.$ Then by equation (3.18), $h_{\nu '}(\Phi ){\kern-1pt}={\kern-1pt}0.$ By equations (3.17) and (3.18), we have $d^*(\nu ',\mu _0)<\zeta .$ Now by equation (3.10), we choose $\theta \in [0,1]$ such that $\nu =\theta \mu _0+(1-\theta )\nu '$ satisfies $P(\Phi ,\chi ,\nu )=c.$ Then by Lemma 3.13(1),

(3.19) $$ \begin{align} \mathcal{P}_{A,B}(\nu)=\alpha \quad\text{and}\quad d^*(\nu,\mu_0)<\zeta. \end{align} $$

So $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )= c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ Since $\Phi $ is expansive on $\Lambda ,$ the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Hence, $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):P(\Phi ,\chi ,\mu )\in [c,c+{1}/{n})\}$ is open in $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):P(\Phi ,\chi ,\mu )\geq c\}$ for any $n\in \mathbb {N^{+}}.$ Then

$$ \begin{align*} &\bigg\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\in\bigg[c,c+\frac{1}{n}\bigg)\bigg\}\\&\quad \text{ is open and dense in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}, \end{align*} $$

and thus $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )=c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi , \chi ,\mu ) \geq c\}.$

3.5.2 Proof of Theorem 3.10

(1) Fix $\alpha {\kern-1pt}\in{\kern-1pt} \mathrm {Int}(L_{A,B})$ , $\mu _0{\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha )$ , $\max _{\mu \in M_{A,B}(\alpha )} \alpha (\mu ) \leq c\leq P(\Phi ,\chi ,\mu _0)$ , and $\eta , \zeta>0.$ By Lemma 3.18(3), there exists $\nu '\in M_{A,B}(\alpha )$ such that $P(\Phi ,\chi ,\nu ')=c$ and $d^*(\nu ',\mu _0)< {\zeta }/{2}.$ For $\alpha \in \mathrm {Int}(L_{A,B})$ , $\nu '\in M_{A,B}(\alpha )$ , and $\eta , {\zeta }/{2}>0,$ there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\nu ') < {\zeta }/{2}$ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\nu ')|<\eta .$ Then we complete the proof of item (1).

(2) Fix $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c<H_{A,B}(\Phi ,\chi ,\alpha ).$ First we show that

(3.20) $$ \begin{align} \{\mu\in M_{A,B}^{\mathrm{erg}}(\alpha):P(f,\alpha,\mu)\geq c\} \text{ is dense in }\{\mu\in M_{A,B}(\alpha):P(f,\alpha,\mu)\geq c\}. \end{align} $$

Let $\mu _0\in M_{A,B}(\alpha )$ be an invariant measure with $P(\Phi ,\chi ,\mu _0) \geq c$ and $\zeta>0$ . If $P(\Phi ,\chi ,\mu _0)>c$ , then there is $\eta>0$ such that $c<c+\eta <P(\Phi ,\chi ,\mu _0).$ For $\alpha \in \mathrm {Int}(L_{A,B})$ , $\mu _0\in M_{A,B}(\alpha )$ , and $\eta , \zeta>0,$ there exists an ergodic measure $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$ If $P(\Phi ,\chi ,\mu _0)=c$ , then we can pick an invariant measure $\mu '\in M_{A,B}(\alpha )$ such that $c<P(\Phi ,\chi ,\mu ') \leq H_{A,B}(\Phi ,\chi ,\alpha )$ , and next pick a sufficiently small number $\theta \in (0,1)$ such that $d^*(\mu _0, \mu ")<\zeta / 2$ , where $\mu "=(1-\theta ) \mu _0+\theta \mu '.$ By equation (3.10), we have $P(\Phi ,\chi ,\mu ")>c.$ By the same argument, there exists an ergodic measure $\nu \in M_{A,B}^{\mathrm {erg}}y(\alpha )$ such that $d^*(\nu ,\mu ")<\zeta /2$ and $P(\Phi ,\chi ,\nu )> c.$ So $d^*(\nu ,\mu _0)<\zeta .$

By equation (3.20) and Lemma 3.18(1),

(3.21) $$ \begin{align} \{\mu\in M_{A,B}^{\mathrm{erg}}(\alpha):P(\Phi,\chi,\mu)\geq c\}\text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

By Lemma 3.18(3),

(3.22) $$ \begin{align} \{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)=c\} \text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

If there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ , then by Lemma 3.18(2), we have

(3.23) $$ \begin{align} \{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c,\ S_{\mu}=\Lambda\}\text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

So by equations (3.21), (3.22), and (3.23), we complete the proof of item (2).

(3) Fix $\alpha \in \mathrm {Int}(L_{A,B})$ and $\mu _0\in M_{A,B}(\alpha )$ with $\max \nolimits _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu _0)< H_{A,B}(\Phi ,\chi ,\alpha ).$ Then by item (2), the set $\{\mu \in M_{A,B}^{\mathrm {erg}}y(\alpha ):P(\Phi ,\chi ,\mu )=P(\Phi ,\chi ,\mu _0)\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq P(\Phi ,\chi ,\mu _0)\}.$ In particular, there is $\mu _{\alpha }\in M_{A,B}^{\mathrm {erg}}(\alpha )$ so that $P(\Phi ,\chi ,\mu _{\alpha })=P(\Phi ,\chi ,\mu _0).$

4 Proofs of Theorems A and B

Now we use ‘multi-horseshoe’ dense property and the results of asymptotically additive families obtained in §§2 and 3 to give a more general result than Theorems A and B.

Theorem 4.1. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a basic set. Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function satisfying equations (3.10) and (3.11). Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Then:

1. (i) for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu _0\in M_{A,B}(\alpha ),$ any $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c\leq P(\phi ,\chi ,\mu _0)$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-c|<\eta ;$

2. (ii) for any $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg}}y(\alpha ):P(\Phi ,\chi ,\mu )=c,\ S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\};$

3. (iii) the set $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu )):\mu \in \mathcal {M}(\Phi ,{\kern-1pt}\Lambda ),\ \alpha {\kern-1pt}={\kern-1pt}\mathcal {P}_{A,B}(\mu ){\kern-1pt}\in{\kern-1pt} \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )} \chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}(\Phi ,\chi ,\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu {\kern-1pt}\in{\kern-1pt} \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ),\ \alpha {\kern-1pt}={\kern-1pt}\mathcal {P}_{A,B}(\mu ){\kern-1pt}\in{\kern-1pt} \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}(\Phi ,\chi ,\alpha )\}.$

If further $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4), $a^i, b^i$ has bounded variation and $\sup _{t \in [0, s]}\|a^i_{t}\|_{\infty }<\infty $ , $\sup _{t \in [0, s]}\|b^i_{t}\|_{\infty }<\infty $ for some $s>0$ and for any $1\leq i\leq d.$ Then:

1. (iv) the following two set are equal

   $$ \begin{align*}\{(\mathcal{P}_{A,B}(\mu), h_{\mu}(\Phi)):\mu\in \mathcal{M}(\Phi,\Lambda),\ \mathcal{P}_{A,B}(\mu)\in\mathrm{Int}(L_{A,B})\},\end{align*} $$
   $$ \begin{align*}\{(\mathcal{P}_{A,B}(\mu), h_{\mu}(\Phi)):\mu\in \mathcal{M}_{\mathrm{erg}}yy(\Phi,\Lambda),\ \mathcal{P}_{A,B}(\mu)\in\mathrm{Int}(L_{A,B})\}.\end{align*} $$

Proof. It is known that each basic set is expansive, there is an invariant measure with full support, the set of periodic measures supported on in $\Lambda $ is a dense subset of $\mathcal {M}(\Phi ,\Lambda )$ , and thus $\{\mu \in \mathcal {M}(\Phi ,M):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,M)$ . Then we obtain items (i)–(iii) by Theorems 2.5, 3.7, 3.9, and 3.10.

Let $\chi =0,$ then by item (iii), we have $\{(\mathcal {P}_{A,B}(\mu ), h_{\mu }(\Phi )):\mu \in \mathcal {M}(\Phi ,\Lambda ),\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}),\ 0\leq h_{\mu }(\Phi )< H_{A,B}(\Phi ,\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), h_{\mu }(\Phi )):\mu \in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ),\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}),\ 0\leq h_{\mu }(\Phi )< H_{A,B}(\Phi ,\alpha )\}.$ Combining with Theorems 3.4 and 3.5, we obtain item (iv).

Now we give the proofs of Theorems A and B.

For a continuous function g, let $g_{t}=\int _{0}^{t} g(\phi _{\tau }(x))\,d\tau $ , then $\sup _{t \in [0, 1]}\|g_{t}\|_{\infty }<\infty $ , $(g_t)_{t\geq 0}$ is an additive family of continuous functions. So letting $\chi \equiv 0$ and $d=1$ in Theorem 4.1(ii), we obtain Theorem A.

Let $\chi \equiv 0$ and $d=2$ in Theorem 4.1(ii) for any $\alpha \in \mathrm {Int}(L_{g,h})$ and

$$ \begin{align*}0\leq c< \max\{h_{\nu}(X):\nu\in M_{g,h}(\alpha)\},\end{align*} $$

the set $\{\mu \in M_{g,h}^{\mathrm {erg}}y(\alpha ):h_{\mu }(X)=c,\ S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{g,h}(\alpha ):h_{\mu }(X)\geq c\}.$ Take $c=0,$ then the set $\{\mu \in M_{g,h}^{\mathrm {erg}}(\alpha ):S_{\mu }=\Lambda \}$ is dense in $M_{g,h}(\alpha ).$ From [14, Proposition 5.7], $\mathcal {M}_{\mathrm {erg}}yy(\Phi ,\Lambda )$ is a $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda ).$ Then $M_{g,h}^{\mathrm {erg}}(\alpha )$ is a $G_{\delta }$ subset of $M_{g,h}(\alpha ),$ and thus $M_{g,h}^{\mathrm {erg}}y(\alpha )$ is residual in $M_{g,h}(\alpha ).$ Since $M_{g,h}(\alpha )$ is convex, by Lemma 3.17, the set $\{\mu \in M_{g,h}(\alpha ):S_{\mu }=\Lambda \}$ is residual in $M_{g,h}(\alpha ).$ So $\{\mu \in M_{g,h}^{\mathrm {erg}}(\alpha ):S_{\mu }=\Lambda \}$ is residual in $M_{g,h}(\alpha )$ and we obtain Theorem B.

5 Singular hyperbolic attractors

In this section, we consider singular hyperbolic attractors and give corresponding results on Question 1.2.

5.1 Singular hyperbolicity and geometric Lorenz attractors

First, we recall the definition of singular hyperbolicity which was introduced by Morales, Pacifico, and Pujals [29] to describe the geometric structure of Lorenz attractors and these ideas were extended to higher dimensional cases in [25, 28].

Definition 5.1. Given a vector field $X\in \mathscr {X}^1(M)$ , a compact and invariant set $\Lambda $ is singular hyperbolic if it admits a continuous $\mathrm {D}\phi _t$ -invariant splitting $T_{\Lambda }M=E^{ss}\oplus E^{cu}$ and constants $C,\eta>0$ such that, for any $x\in \Lambda $ and any $t\geq 0$ :

• • $E^{ss}\oplus E^{cu}$ is a dominated splitting: $\|\mathrm {D}\phi _t|_{E^{ss}(x)}\|\cdot \|\mathrm {D}\phi _{-t}|_{E^{cu}(\phi _t(x))}\|< Ce^{-\eta t}$ ;

• • $E^{ss}$ is uniformly contracted by $\mathrm {D}\phi _t$ : $\|\mathrm {D}\phi _t(v)\|< Ce^{-\eta t}\|v\|$ for any $v\in E^{ss}(x)\setminus \{0\}$ ;

• • $E^{cu}$ is sectionally expanded by $\mathrm {D}\phi _t$ : $|\det \mathrm {D}\phi _t|_{V_x}|> Ce^{\eta t}$ for any two-dimensional subspace $V_x\subset E^{cu}_x$ .

Lemma 5.2. [30, Theorem A and Lemma 2.9]

Given a vector field $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , if $\Lambda $ is sectional hyperbolic, all the singularities in $\Lambda $ are hyperbolic, then $\Phi $ is entropy expansive on $\Lambda $ and thus the metric entropy function $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous.

We recall the concept of a homoclinic class of a hyperbolic periodic orbit.

Definition 5.3. Given a vector field $X\in \mathscr {X}^1(M)$ , an invariant compact subset $\Lambda \subset M$ is a homoclinic class if there exists a hyperbolic periodic point $p\in \Lambda \cap \text {Per}(X)$ so that

$$ \begin{align*} \Lambda\colon=\overline{W^{s}(\operatorname{Orb}(p)) \pitchfork W^{u}(\operatorname{Orb}(p))}, \end{align*} $$

that is, it is the closure of the points of transversal intersection between stable and unstable manifolds of the periodic orbit $\operatorname {Orb}(p)$ of p. We say a homoclinic class is non-trivial if it is not reduced to a single hyperbolic periodic orbit.

From [2, Theorem 2.17], any non-empty homoclinic class $\Lambda $ contains a dense set of periodic orbits. Then there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ by [14, Proposition 21.12].

Lemma 5.4. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a homoclinic class of X. Then there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda .$

Now we give the definition of geometric Lorenz attractors following Guckenheimer and Williams [18, 19, 39] for vector fields on a closed 3-manifold $M^3$ .

Definition 5.5. We say $X\in \mathscr {X}^r(M^3)$ ( $r\geq 1$ ) exhibits a geometric Lorenz attractor $\Lambda $ if X has an attracting region $U\subset M^3$ such that $\Lambda =\bigcap _{t>0}\phi ^X_t(U)$ is a singular hyperbolic attractor and satisfies the following (see Figure 2).

• • There exists a unique singularity $p\in \Lambda $ with three exponents $\unicode{x3bb} _1<\unicode{x3bb} _2<0<\unicode{x3bb} _3$ , which satisfy $\unicode{x3bb} _1+\unicode{x3bb} _3<0$ and $\unicode{x3bb} _2+\unicode{x3bb} _3>0$ .

• • $\Lambda $ admits a $C^r$ -smooth cross-section S which is $C^1$ -diffeomorphic to $[-1,1]\times [-1,1]$ such that $\Gamma =\{0\}\times [-1,1]=W^s_{\mathrm {loc}}(\sigma )\cap S$ , and for every $z\in U\setminus W^s_{\mathrm {loc}}(\sigma )$ , there exists $t>0$ such that $\phi _t^X(z)\in S$ .

• • Up to the previous identification, the Poincaré map $P:S\setminus \Gamma \rightarrow S$ is a skew-product map

  $$ \begin{align*} P(x,y)=\big( f(x)~,~g(x,y) \big)\quad\text{for all }(x,y)\in[-1,1]^2\setminus \Gamma. \end{align*} $$
  Moreover, it satisfies:

  1. – $g(x,y)<0$ for $x>0$ , and $g(x,y)>0$ for $x<0$ ;

  2. – $\sup _{(x,y)\in S\setminus \Gamma }\big |\partial g(x,y)/\partial y\big |<1$ and $\sup _{(x,y)\in S\setminus \Gamma }\big |\partial g(x,y)/\partial x\big |<1$ ;

  3. – the one-dimensional quotient map $f:[-1,1]\setminus \{0\}\rightarrow [-1,1]$ is $C^1$ -smooth and satisfies $\lim _{x\rightarrow 0^-}f(x)=1$ , $\lim _{x\rightarrow 0^+}f(x)=-1$ , $-1<f(x)<1$ , and $f'(x)>\sqrt {2}$ for every $x\in [-1,1]\setminus \{0\}$ .

Figure 2 Geometric Lorenz attractor and return map.

It has been proved that the geometric Lorenz attractor is a homoclinic class [2, Theorem 6.8] and $C^2$ -robust [34, Proposition 4.7].

Proposition 5.6. Let $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ and $X\in \mathscr {X}^r(M^3)$ . If X exhibits a geometric Lorenz attractor $\Lambda $ with attracting region U, then there exists a $C^r$ -neighborhood $\mathcal {U}$ of X in $\mathscr {X}^r(M^3)$ such that for every $Y\in \mathcal {U}$ , U is an attracting region of Y, and the maximal invariant set $\Lambda _Y=\bigcap _{t>0}\phi _t^Y(U)$ is a geometric Lorenz attractor. Moreover, the geometric Lorenz attractor is a singular hyperbolic homoclinic class, and every pair of periodic orbits are homoclinic related.

For singular hyperbolic attractors, we have the following result.

Theorem C. There exists a Baire residual subset $\mathcal {R}^r\subset \mathscr {X}^r(M^3),(r\in \mathbb {N}_{\geq 2})$ and a Baire residual subset $\mathcal {R}\subset \mathscr {X}^1(M)$ so that if $\Lambda $ is a geometric Lorenz attractor of $X\in \mathcal {R}^r$ or a singular hyperbolic attractor of $X\in \mathcal {R},$ then for any continuous function g, h on $\Lambda $ , we have:

1. (i) for any $\alpha \in \mathrm {Int}(L_{g})$ and $0\leq c< \max \{h_{\mu }(\Phi ):\mu \in M_{g}(\alpha )\},$ the set $\{\mu \in M_{g}^{\mathrm {erg}}y(\alpha ):h_{\mu }(\Phi )=c,\ S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{g}(\alpha ):h_{\mu }(\Phi )\geq c\};$

2. (ii) for any $\alpha \in \mathrm {Int}(L_{g,h}),$ the set $\{\mu \in M_{g,h}^{\mathrm {\mathrm {erg}}}(\alpha ):S_{\mu }=\Lambda \}$ is residual in $M_{g,h}(\alpha ).$

Remark 5.7. We will prove one general result Theorem 5.15 stating that when $\Lambda $ is a singular hyperbolic homoclinic class such that each pair of periodic orbits are homoclinically related and $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}=\overline {\mathcal {M}_{\mathrm {per}}(\Lambda )}$ , then the conclusions of Theorem C hold. Then Theorem C is a direct consequence of Theorem 5.15. The reason is that when $\Lambda $ is a Lorenz attractor of vector fields in a Baire residual subset $\mathcal {R}^r\subset \mathcal {X}^r(M^3), (r\in \mathbb {N}_{\geq 2})$ or $\Lambda $ is a singular hyperbolic attractor $\Lambda $ of vector fields in a Baire residual set $\mathcal {R}\subset \mathcal {X}^1(M)$ , then $\Lambda $ is a homoclinic class such that each pair of periodic orbits are homoclinically related (cf. [2, Theorem 6.8] for Lorenz attractors and [12, Theorem B] for singular hyperbolic attractors) and $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}=\overline {\mathcal {M}_{\mathrm {per}}(\Lambda )}$ (cf. [34, Theorems A and B]).

5.2 Proof of Theorem C

Given a vector field $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , let $\mathcal {N}\subset \mathcal {M}(\Phi ,\Lambda )$ be a convex set. We denote $\mathcal {N}_{\mathrm {erg}}y=\mathcal {M}_{\mathrm {\mathrm {erg}}}(\Phi ,\Lambda )\cap \mathcal {N}.$ Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4). Denote

$$ \begin{align*}L_{A,B}^{\mathcal{N}}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{N}\}.\end{align*} $$

For any $\alpha \in L_{A,B}^{\mathcal {N}},$ denote

$$ \begin{align*}M_{A,B}^{\mathcal{N}}(\alpha)=\{\mu\in {\mathcal{N}}:\mathcal{P}_{A,B}(\mu)=\alpha\},\quad M_{A,B}^{\mathrm{erg},\mathcal{N}}(\alpha)=\{\mu\in \mathcal{N}_{\mathrm{erg}}y:\mathcal{P}_{A,B}(\mu)=\alpha\}.\end{align*} $$

Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. We define the pressure of $\chi $ with respect to $\mu $ by $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\chi (\mu ).$ For any $\alpha \in L_{A,B}^{\mathcal {N}},$ denote

$$ \begin{align*}H_{A,B}^{\mathcal{N}}(\Phi,\chi,\alpha)=\sup\{P(\phi,\chi,\mu):\mu\in M_{A,B}^{\mathcal{N}}(\alpha)\}.\end{align*} $$

In particular, when $\chi \equiv 0,$ we write

$$ \begin{align*}H_{A,B}^{\mathcal{N}}(\Phi,\alpha)=H_{A,B}^{\mathcal{N}}(\Phi,0,\alpha)=\sup\{h_{\mu}(\Phi):\mu\in M_{A,B}^{\mathcal{N}}(\alpha)\}.\end{align*} $$

If we replace $\mathcal {M}(\Phi ,\Lambda )$ by $\mathcal {N}$ in §§3.3–3.5, the results of Theorems 3.9, 3.10, and Remark 3.12 also hold. In fact, the convexity of $\mathcal {M}(\Phi ,\Lambda )$ is one core property in the proof of Theorems 3.9, 3.10, and Remark 3.12. Since $\mathcal {N}$ is convex, then the arguments of §3.3 are also true. Here, we omit the proof.

Theorem 5.8. Given a vector field $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , assume that the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4). Let $\mathcal {N}\subset \mathcal {M}(\Phi ,\Lambda )$ be a convex set. Assume that the following holds: for any $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {N},$ and any $\eta , \zeta>0$ , there are compact invariant subsets $\Lambda _i\subseteq \Theta \subsetneq \Lambda $ such that for each $i\in \{1,2,\ldots ,m\}$ :

1. (1) for any $a\in \mathrm {Int}(\mathcal {P}_{A,B}(\mathcal {M}(\Phi ,\Theta )))$ and any $\varepsilon>0,$ there exists an ergodic measure $\mu _a\in \mathcal {N}$ supported on $\Theta $ with $\mathcal {P}_{A,B}(\mu _a)=a$ such that $|h_{\mu _a}(\Phi )-H(\Phi ,a,\Theta )|<\varepsilon ,$ where $H(\Phi ,a,\Theta )=\sup \{h_{\mu }(\Phi ):\mu \in \mathcal {M}(\Phi ,\Theta ) \text { and }\mathcal {P}_{A,B}(\mu )=a\};$

2. (2) $h_{\mathrm {top}}(\Lambda _i)>h_{\mu _i}(\Phi )-\eta ;$

3. (3) $d_H(K, \mathcal {M}(\Phi , \Theta ))<\zeta $ , $d_H(\mu _i, \mathcal {M}(\Phi , \Lambda _i))<\zeta .$

Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. Then we have the following.

1. (i) For any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {N}}),$ any $\mu _0\in M_{A,B}^{\mathcal {N}}(\alpha )$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg},\mathcal {N}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$

If further $\{\mu \in \mathcal {N}:h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {N},$ and $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ satisfies equations (3.10) and (3.11), then we have:

1. (ii) for any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {N}}),$ any $\mu _0\in M_{A,B}^{\mathcal {N}}(\alpha ),$ any $\max _{\mu \in M_{A,B}^{\mathcal {N}}(\alpha )}\chi (\mu )\leq c\leq P(\phi ,\chi ,\mu _0)$ and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg},\mathcal {N}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-c|<\eta ;$

2. (iii) for any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {N}})$ and $\max _{\mu \in M_{A,B}^{\mathcal {N}}(\alpha )}\chi (\mu )\leq c< H_{A,B}^{\mathcal {N}}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg},\mathcal {N}}(\alpha ):P(\Phi ,\chi ,\mu )=c\}$ is residual in $\{\mu \in M_{A,B}^{\mathcal {N}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ If further there is an invariant measure $\tilde {\mu }\in \mathcal {N}$ with $S_{\tilde {\mu }}=\Lambda $ , then for any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {N}})$ and $\max _{\mu \in M_{A,B}^{\mathcal {N}}(\alpha )}\chi (\mu )\leq c< H_{A,B}^{\mathcal {N}}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg},\mathcal {N}}(\alpha ):P(\Phi ,\chi ,\mu )=c,\ S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{A,B}^{\mathcal {N}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\};$

3. (iv) the set $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )) : \mu \in \mathcal {N},\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}^{\mathcal {N}}), \max _{\mu \in M_{A,B}^{\mathcal {N}} (\alpha )} \chi (\mu ) \leq P(\Phi ,\chi ,\mu )< H_{A,B}^{\mathcal {N}}(\Phi ,\chi ,\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu \in \mathcal {N}_{\mathrm {erg}},\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}^{\mathcal {N}}), \max _{\mu \in M_{A,B}^{\mathcal {N}}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}^{\mathcal {N}} (\Phi ,\chi ,\alpha )\}.$

Denote by $\operatorname {Sing}(\Lambda )$ the set of singularities for the vector field X in $\Lambda $ , by $\mathcal {M}_{\mathrm {per}}(\Lambda )$ the set of periodic measures supported on $\Lambda $ , and set

$$ \begin{align*}\mathcal{M}_1(\Lambda)=\{\mu\in \mathcal{M}(\Phi,\Lambda): \mu(\operatorname{Sing}(\Lambda))=0 \}\quad\text{and}\quad\mathcal{M}_0(\Lambda)=\mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)\cap \mathcal{M}_1(\Lambda).\end{align*} $$

Proposition 5.9. [33, Proposition 4.12]

Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a singular hyperbolic homoclinic class of X. Assume each pair of periodic orbits of $\Lambda $ are homoclinic related. Then for each $\varepsilon>0$ and any $\mu \in \mathcal {M}_1(\Lambda )$ , there exist a basic set $\Lambda '\subset \Lambda $ and $\nu \in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ')$ so that

$$ \begin{align*} d^*(\nu,\mu)<\varepsilon \quad\text{and}\quad h_{\nu}(\Phi)>h_{\mu}(\Phi)-\varepsilon. \end{align*} $$

Proposition 5.10. [33, Proposition 4.13]

Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a singular hyperbolic homoclinic class of X. Assume each pair of periodic orbits of $\Lambda $ are homoclinic related, and $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}$ . Then for each $\varepsilon>0$ and any $\mu \in \mathcal {M}(\Phi ,\Lambda )$ , there exist a basic set $\Lambda '\subset \Lambda $ and $\nu \in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ')$ so that

$$ \begin{align*} d^*(\nu,\mu)<\varepsilon ~~~\text{and}~~~h_{\nu}(\Phi)>h_{\mu}(\Phi)-\varepsilon. \end{align*} $$

Lemma 5.11. [33, Lemma 4.5]

Let $\Lambda _1$ and $\Lambda _2$ be two basic sets of $X\in \mathcal {X}^1(M)$ . Assume there exists hyperbolic periodic points $p_1\in \Lambda _1$ and $p_2\in \Lambda _2$ such that $\operatorname {Orb}(p_1)$ and $\operatorname {Orb}(p_2)$ are homoclinically related. Then there exists a larger basic set $\Lambda $ that contains both $\Lambda _1$ and $\Lambda _2$ .

Now we state a result on the ‘multi-horseshoe’ dense property of singular hyperbolic homoclinic classes.

Theorem 5.12. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a singular hyperbolic homoclinic class of X. Assume each pair of periodic orbits of $\Lambda $ are homoclinic related. Then $\Lambda $ satisfies the ‘multi-horseshoe’ dense property on $\mathcal {M}_1(\Lambda )$ . Moreover, if $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}=\overline {\mathcal {M}_{\mathrm {per}}(\Lambda )}$ , then $\Lambda $ satisfies the ‘multi-horseshoe’ dense property.

Proof. Fix $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}_1(\Lambda )(\mathcal {M}(\Phi ,\Lambda ))$ and $\eta , \zeta>0.$ By Proposition 5.9 (Proposition 5.10), for each $1\leq i\leq m$ , there exist a basic set $\Lambda ^{\prime }_i\subset \Lambda $ and $\nu _i\in \mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda ^{\prime }_i)$ so that

$$ \begin{align*} d^*(\nu_i,\mu_i)<\frac{\zeta}{2} ~~~\text{and}~~~h_{\nu_i}(\Phi)>h_{\mu_i}(\Phi)-\frac{\eta}{2}. \end{align*} $$

By Lemma 5.11, there exists a larger basic set $\tilde {\Lambda }$ that contains every $\Lambda ^{\prime }_i$ . Applying Theorem 2.5 to $\tilde {\Lambda }$ , $\tilde {F}=\operatorname {\mathrm {cov}}\{\nu _i\}_{i=1}^m\subseteq \mathcal {M}(\Phi ,\tilde {\Lambda })$ and ${\eta }/{2}$ , ${\zeta }/{2}$ , we complete the proof.

Now we show that the results of Theorem 5.8 hold for singular hyperbolic homoclinic classes.

Theorem 5.13. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a singular hyperbolic homoclinic class of X. Assume each pair of periodic orbits of $\Lambda $ are homoclinic related. Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function satisfying equations (3.10) and (3.11). Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Then:

1. (i) for any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {M}_1(\Lambda )}),$ any $\mu _0\in M_{A,B}^{\mathcal {M}_1(\Lambda )}(\alpha ),$ any $\max _{\mu \in M_{A,B}^{\mathcal {M}_1(\Lambda )}(\alpha )}\chi (\mu )\leq c\leq P(\phi ,\chi ,\mu _0)$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg},{\mathcal {M}_1(\Lambda )}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-c|<\eta ;$

2. (ii) for any $\alpha \in \mathrm {Int}(L_{A,B}^{\mathcal {M}_1(\Lambda )})$ and $\max _{\mu \in M_{A,B}^{\mathcal {M}_1(\Lambda )}(\alpha )}\chi (\mu )\leq c< H_{A,B}^{\mathcal {M}_1(\Lambda )}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg},{\mathcal {M}_1(\Lambda )}}(\alpha ):P(\Phi ,\chi ,\mu )=c\}$ is residual in $\{\mu \in M_{A,B}^{\mathcal {M}_1(\Lambda )}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\};$

3. (iii) the set $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu \in {\mathcal {M}_1(\Lambda )},\ \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}^{\mathcal {M}_1(\Lambda )}), \max _{\mu \in M_{A,B}^{\mathcal {M}_1(\Lambda )}}{\kern-1pt} {{}_{_{(\alpha )}}}\chi (\mu ){\kern-1pt}\leq{\kern-1pt} P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ){\kern-1.5pt}<{\kern-2.5pt} H_{A,B}^{\mathcal {M}_1(\Lambda )}(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ))\kern-1pt:\kern-1pt \mu {\kern-1pt}\in{\kern-1pt} {\mathcal {M}_0(\Lambda )},\ \alpha {\kern-1pt}={\kern-1pt}\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}^{\mathcal {M}_1(\Lambda )}), \max _{\mu \in M_{A,B}^{\mathcal {M}_1(\Lambda )}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}^{\mathcal {M}_1(\Lambda )}(\Phi ,\chi ,\alpha )\}.$

Proof. Note that since $\Lambda $ is a singular hyperbolic homoclinic class, the vector field X satisfies the star condition in a neighborhood of $\Lambda $ . More precisely, there exist a neighborhood U of $\Lambda $ and a $C^1$ -neighborhood $\mathcal {U}$ of X in $\mathscr {X}^1(M)$ such that every critical element contained in U associated to a vector field $Y\in \mathcal {U}$ is hyperbolic. Then by Lemma 5.2, the entropy function is upper semi-continuous. Note that $\mathcal {M}(\Phi ,\Theta )\subset \mathcal {M}_1(\Lambda )$ if $\Theta \subset \Lambda $ is a horseshoe. Since the metric entropy of periodic measure is zero, then $\{\mu \in \mathcal {M}_1(\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $ \mathcal {M}_1(\Lambda )$ by Theorem 5.12. So we complete the proof by Theorems 3.7, 5.8, and 5.12.

Combining with Theorem 5.6, the results of Theorem 5.13 hold for geometric Lorenz attractors.

Corollary 5.14. Let $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ and $X\in \mathscr {X}^r(M^3)$ . If X exhibits a geometric Lorenz attractor $\Lambda $ , then the results of Theorem 5.13 hold for $\Lambda .$

If further we have $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}=\overline {\mathcal {M}_{\mathrm {per}}(\Lambda )}$ , then using Theorems 3.7, 3.9, 3.10, Remark 3.12, Theorem 5.12, and Lemma 5.4, we have the following result for singular hyperbolic homoclinic classes.

Theorem 5.15. Let $X\in \mathscr {X}^1(M)$ and $\Lambda $ be a singular hyperbolic homoclinic class of X. Assume each pair of periodic orbits of $\Lambda $ are homoclinic related, and $\mathcal {M}(\Phi ,\Lambda )=\overline {\mathcal {M}_1(\Lambda )}=\overline {\mathcal {M}_{\mathrm {per}}(\Lambda )}$ . Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function satisfying equations (3.10) and (3.11). Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Then:

1. (i) for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu _0\in M_{A,B}(\alpha ),$ any $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c\leq P(\phi ,\chi ,\mu _0)$ , and any $\eta , \zeta>0$ , there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-c|<\eta ;$

2. (ii) for any $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the set $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )=c,\ S_{\mu }{\kern-1pt}={\kern-1pt}\Lambda \}$ is residual in $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt} \mu ){\kern-1pt}\geq{\kern-1pt} c\};$

3. (iii) the set $\{(\mathcal {P}_{A,B}(\mu ),{\kern-1pt} P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu )):\mu {\kern-1pt}\in{\kern-1pt} \mathcal {M}(\Phi ,M), \alpha =\mathcal {P}_{A,B}(\mu )\in \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )} \chi (\mu )\leq P(\Phi ,\chi ,\mu )< H_{A,B}(\Phi ,\chi ,\alpha )\}$ coincides with $\{(\mathcal {P}_{A,B}(\mu ), P(\Phi ,\chi ,\mu )):\mu {\kern-1pt}\in{\kern-1pt} \mathcal {M}_{\mathrm {erg}}(\Phi ,M),{\kern-1pt}\ \alpha =\mathcal {P}_{A,B}(\mu ){\kern-1pt}\in{\kern-1pt} \mathrm {Int}(L_{A,B}), \max _{\mu \in M_{A,B}(\alpha )}\chi (\mu ){\kern-1pt}\leq P(\Phi ,\chi ,\mu ){\kern-1pt}< H_{A,B}(\Phi ,\chi ,\alpha )\}.$

Proceeding in a similar manner to Theorems A and B, letting $\chi \equiv 0$ and $d=1,2$ in Theorem 5.15, we obtain Theorem C by Remark 5.7.