Relative pressure functions and their equilibrium states

Abstract

For a subshift $(X, \sigma _{X})$ and a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log f_{n}\, d\kern-1pt\mu =\int h \,d\kern-1pt\mu $ for every invariant measure $\mu $ on X. For this purpose, we first we study necessary and sufficient conditions for ${\mathcal F}$ to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map $\pi : X\rightarrow Y$ , where $(X, \sigma _{X})$ is an irreducible shift of finite type and $(Y, \sigma _{Y})$ is a subshift, applying our results and the results obtained by Cuneo [Additive, almost additive and asymptotically additive potential sequences are equivalent. Comm. Math. Phys. 37 (3) (2020), 2579–2595] on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application, we study the projection $\pi \mu $ of an invariant weak Gibbs measure $\mu $ for a continuous function on an irreducible shift of finite type.

1 Introduction

The thermodynamic formalism for sequences of continuous functions generalizes the formalism for continuous functions and has been applied to solve some dimension problems in non-conformal dynamical systems. The equilibrium states for sequences of continuous functions are the equilibrium states for Borel measurable functions in general. In [10] Falconer introduced the thermodynamic formalism for subadditive sequences to study repellers of non-conformal transformations. Cao, Feng and Huang in [6] established the theory for subadditive sequences wherein the variational principle was obtained for compact dynamical systems. Asymptotically additive sequences, which generalize the almost additive sequences studied by Barreira [2] and Mummert [18], were also introduced by Feng and Huang [13]. The properties of equilibrium states for sequences of continuous functions, such as uniqueness, the (generalized) Gibbs property and mixing properties, have been also studied (see, for example, [2, 12, 18]). Here, a natural question arises.

Question 1. Given a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on a compact metric space X, what are necessary and sufficient conditions for the existence of a continuous function h on X such that

(1) $$ \begin{align} \lim_{n\rightarrow\infty}\frac{1}{n}\int \log f_{n} \,d\kern-1pt\mu=\int h \,d\kern-1pt\mu \end{align} $$

for every invariant Borel probability measure $\mu $ on X?

If such an h exists, then the thermodynamic formalism for such sequences ${\mathcal F}$ reduces to the formalism for continuous functions. Cuneo [9, Theorem 1.2] proved that if a sequence of continuous functions is asymptotically additive (see (4) for the definition), then there always exists $h\in C(X)$ satisfying (1) for every invariant measure $\mu $ on X. In this paper, we study necessary conditions for a subadditive sequence ${\mathcal F}$ on an irreducible subshift $(X, \sigma _{X})$ to have a continuous function $h \in C(X)$ satisfying (1) for every invariant measure $\mu $ on X. Using our results and the result obtained by Cuneo [9, Theorem 1.2], we give some answers to Question 1 (Theorems 4.3, 6.9 and 7.8). Towards this end, we first study conditions for a subadditive sequence on a subshift to be an asymptotically additive sequence in terms of certain properties for periodic points. Given a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X, if (1) holds for every invariant Borel probability measure $\mu $ on X, then the sequence $\tilde {\mathcal F}=\{(1/n)\log (f_{n}/e^{S_{n}h})\}_{n=1}^{\infty }$ converges (pointwise) to the zero function $0$ for every periodic point of $\sigma _{X}$ (see Proposition 3.1). We show in Theorems 4.3 and 4.4 that if the sequence $\tilde {\mathcal F}$ converges (pointwise) to $0$ for every periodic point of $\sigma _{X}$ and ${\mathcal F}$ satisfies a particular property for certain periodic points then $\tilde {\mathcal F}$ converges to $0$ everywhere; moreover, it converges uniformly to $0$ on X. This gives the asymptotic additivity of ${\mathcal F}$ . We apply Theorem 4.3 when we study Question 1 with regard to a relative pressure function of a continuous function (Theorems 6.9 and 7.8). In Proposition 3.1, Question 1 is studied in a general form. Note that subadditive sequences are not asymptotically additive in general (see Example 7.2 in §7).

In §6, we consider relative pressure functions in relation to compensation functions. Let $(X, \sigma _{X}), (Y, \sigma _{Y})$ be subshifts and $\pi :X\rightarrow Y$ be a factor map. Let $f\in C(X)$ , $n\in {\mathbb N}$ and $\delta>0$ . For each $y\in Y$ , define

$$ \begin{align*}P_{n}(\sigma_{X}, \pi, f, \delta)(y)=\sup\bigg\{\sum_{x\in E}e^{(S_{n}f)(x)}:E \text { is an } (n, \delta) \text{ separated subset of } \pi^{-1}(\{y\})\bigg\},\end{align*} $$

$$ \begin{align*}P(\sigma_{X}, \pi, f, \delta)(y)=\limsup_{n\rightarrow \infty}\frac{1}{n}\log P_{n}(\sigma_{X}, \pi, f,\delta)(y),\end{align*} $$

$$ \begin{align*}P(\sigma_{X}, \pi, f)(y)=\lim_{\delta\rightarrow 0}P(\sigma_{X}, \pi,f, \delta)(y).\end{align*} $$

The function $P(\sigma _{X}, \pi , f):Y\rightarrow {\mathbb R}$ is the relative pressure function of $f\in C(X)$ with respect to $(\sigma _{X}, \sigma _{Y}, \pi )$ . In general it is merely Borel measurable. In Theorem 6.9, for an irreducible shift of finite type $(X, \sigma _{X})$ , we study equivalent conditions for a relative pressure function $P(\sigma _{X}, \pi , f)$ on Y to have a function $h\in C(Y)$ such that

(2) $$ \begin{align} \int P(\sigma_{X}, \pi, f)\,d\kern-1pt\mu=\int h \,d\kern-1pt\mu \quad\textrm{for every } \mu\in M(Y, \sigma_{Y})\end{align} $$

where $M(Y, \sigma _{Y})$ is the set of invariant Borel probability measures on Y. In general, a relative pressure function $P(\sigma _{X}, \pi , f)$ is represented by a subadditive sequence ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ of continuous functions on Y (see (32) for $g_{n}$ ), that is, $P(\sigma _{X}, \pi , f) =\lim _{n\rightarrow \infty }(1/n)\log g_{n}$ almost everywhere with respect to every $\mu \in M(Y, \sigma _{Y})$ . The sequence ${\mathcal G}$ satisfies an additional condition (see (D2) in §2.2) weaker than almost additivity and it is not asymptotically additive in general. We prove that the subadditive sequence ${\mathcal G}$ on Y associated to $P(\sigma _{X}, \pi , f)$ satisfies the particular property for certain periodic points in Lemma 4.1 (ii). Applying Theorem 4.3, we obtain in Theorem 6.9 that, for $h\in C(Y)$ , uniform convergence of $\tilde {\mathcal G}=\{(1/n)\log (g_{n}/e^{S_{n}h})\}_{n=1}^{\infty }$ to 0 on Y is equivalent to pointwise convergence of $\tilde {\mathcal G}$ to $0$ for every periodic point of $\sigma _{Y}$ . In particular, we obtain that (2) holds if and only if the sequence ${\mathcal G}$ associated to $P(\sigma _{X}, \pi , f)$ is asymptotically additive. Moreover, if there exists an invariant weak Gibbs measure m for $f\in C(X)$ , then (2) holds if and only if $\pi m$ is an invariant weak Gibbs measure for some continuous function on Y (Theorem 7.8). The properties of the sequence ${\mathcal G}$ associated to $P(\sigma _{X}, \pi , f)$ under the existence of h in (2) are studied and a condition of non-existence of such a continuous function is also studied (Corollary 6.11). These results are applied directly to study the projection of an invariant weak Gibbs measure for a continuous function on X in §7 (see Theorem 7.6 and Corollary 7.9). Note that in general if there exists an invariant weak Gibbs measure m for $f\in C(X)$ , then $\pi m$ is a weak Gibbs equilibrium state for the subadditive sequence ${\mathcal G}$ associated to $P(\sigma _{X}, \pi , f)$ .

On the other hand, relative pressure functions are connected with compensation functions. Given $f\in C(X)$ , Theorem 6.9 relates the question on the existence of h in (2) with the existence of a compensation function $f-h\circ \pi $ for some $h\in C(Y)$ . A function $F\in C(X)$ is a compensation function for a factor map $\pi $ if

(3) $$ \begin{align} \sup_{\mu\in M(X, \sigma_{X})}\bigg\{h_{\mu}(\sigma_{X})+\int F \,d\kern-1pt\mu+\int \phi \circ \pi \,d\kern-1pt\mu\bigg\}=\sup_{\nu\in M(Y, \sigma_{Y})}\bigg\{h_{\nu}(\sigma_{Y})+\int \phi\, d\nu\bigg\} \end{align} $$

for every $\phi \in C(Y)$ . If $F=G\circ \pi $ , $G\in C(Y)$ , then $G\circ \pi $ is a saturated compensation function. The concept of compensation functions was introduced by Boyle and Tuncel [5], and their properties were studied by Walters [28] in relation to relative pressure. The existence of compensation functions has been studied [1, 24–26]. Shin [25, 26] proved that a saturated compensation function does not always exist and gave a characterization for the existence of a saturated compensation function for factor maps between shifts of finite type. A function $-h\circ \pi \in C(X)$ is a saturated compensation function if and only if (2) holds for $f=0$ . Our results connect the result obtained by Shin with the asymptotic additivity of the sequence associated to $P(\sigma _{X}, \pi , 0)$ (see Remark 6.10 and Corollary 7.1). Since saturated compensation functions were applied to study the measures of full Hausdorff dimension of non-conformal repellers, studying the properties of equilibrium states for h in (2) would help in the further study of certain dimension problems (see Example 7.4).

Section 5 deals with a particular class of subadditive sequences on subshifts satisfying an additional property (see condition (C2)) weaker than almost additivity but stronger than (D2). The result of Feng [12, Theorem 5.5] implies that there is a unique (generalized) Gibbs equilibrium state for a subadditive sequence with bounded variation satisfying property (C2). We study equivalent conditions for this type of sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on a subshift X to have a continuous function for which the unique Gibbs equilibrium state is a weak Gibbs measure (Theorem 5.1). In this case, we obtain that for $h\in C(X)$ uniform convergence of the sequence of functions $\{(1/n)\log (f_{n}/e^{S_{n}h})\}_{n=1}^{\infty }$ to 0 on X is equivalent to pointwise convergence of the sequence to 0 on X. We note that it is not clear that the condition for certain periodic points in Theorem 4.4 (ii) is satisfied for this type of sequence in general.

2 Background

2.1 Shift spaces

We give a brief summary of the basic definitions in symbolic dynamics. $(X, \sigma _{X})$ is a one-sided subshift if X is a closed shift-invariant subset of $\{1,\ldots , k\}^{{\mathbb N}}$ for some $k\geq 1$ , that is, $\sigma _{X}(X)\subseteq X$ , where the shift $\sigma _{X}:X\rightarrow X$ is defined by $(\sigma _{X}(x))_{i}=x_{i+1}$ for all $i\in {\mathbb N}$ , $x=(x_{n})^{\infty }_{n=1} \in X.$ Define a metric d on X by $d(x,x^{\prime })={1}/{2^{k}}$ if $x_{i}=x^{\prime }_{i}$ for all $1\leq i\leq k$ and $x_{k+1}\neq {x^{\prime }}_{k+1}$ , $d(x,x^{\prime })=1$ if $x_{1}\neq x^{\prime }_{1}$ , and $d(x,x^{\prime })=0$ otherwise. Throughout this paper, we consider one-sided subshifts. Define a cylinder set $[x_{1} \ldots x_{n}]$ of length n in X by $[x_{1}\ldots x_{n}]=\{(z_{i})_{i=1}^{\infty } \in X: z_{i}=x_{i} \text { for all }1\leq i\leq n\}.$ For each $n \in {\mathbb N},$ denote by $B_{n}(X)$ the set of all n-blocks that appear in points in X. Define $B_{0}(X)=\{\epsilon \},$ where $\epsilon $ is the empty word of length $0$ . The language of X is the set $B(X)=\bigcup _{n=0}^{\infty }B_{n}(X)$ . A subshift $(X,\sigma _{X})$ is irreducible if for any allowable words $u, v\in B(X)$ , there exists $w\in B(X)$ such that $uwv \in B(X)$ . A subshift has the weak specification property if there exists $p\in {\mathbb N}$ such that for any allowable words $u, v \in B(X)$ , there exist $0\leq k\leq p$ and $w\in B_{k}(X)$ such that $uwv\in B(X)$ . We call such p a weak specification number. A point $x\in X$ is a periodic point of $\sigma _{X}$ if there exists $l\in {\mathbb N}$ such that $\sigma _{X}^{l}(x)=x$ .

Let $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ be subshifts. A shift of finite type $(X,\sigma _{X})$ is one-step if there exists a set F of forbidden blocks of length less than or equal to $2$ such that $X=\{x\in \{1, \ldots , k\}^{{\mathbb N}}: \omega \text{ does not appear in } x \text{ for any }\omega \in F\}$ . A map $\pi :X\rightarrow Y$ is a factor map if it is continuous, surjective and satisfies $\pi \circ \sigma _{X} = \sigma _{Y}\circ \pi $ . If, in addition, the ith position of the image of x under $\pi $ depends only on $x_{i},$ then $\pi $ is a one-block factor map. Throughout the paper we assume that a shift of finite type $(X, \sigma _{X})$ is one-step and $\pi $ is a one-block factor map. Denote by $M(X, \sigma _{X})$ the collection of all $\sigma _{X}$ -invariant Borel probability measures on X and by $\mathrm{Erg}(X, \sigma _{X})$ all ergodic members of $M(X, \sigma _{X})$ .

2.2 Sequences of continuous functions

We give a brief summary on the basic results on the sequences of continuous functions considered in this paper. Let $(X, \sigma _{X})$ be a subshift on finitely many symbols. For each $n\in {\mathbb N}$ , let $f_{n}: X\rightarrow {\mathbb R}^{+}$ be a continuous function. A sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ is almost additive if there exists a constant $C\geq 0$ such that $e^{-C}f_{n}(x) f_{m}(\sigma ^{n}_{X} x) \leq f_{n+m}(x) \leq e^{C}f_{n}(x) f_{m}(\sigma ^{n}_{X} x) $ . In particular, if $C=0$ , then ${\mathcal F}$ is additive. The thermodynamic formalism for almost additive sequences was studied in Barrera [2] and Mummert [18]. More generally, Feng and Huang [13] introduced asymptotically additive sequences which generalize almost additive sequences. A sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ is asymptotically additive on X if for every $\epsilon> 0$ there exists a continuous function $\rho _{\epsilon }$ such that

(4) $$ \begin{align} \limsup_{n\rightarrow \infty} \frac{1}{n} \lVert \log f_{n}-S_{n}\rho_{\epsilon} \rVert_{\infty} < \epsilon, \end{align} $$

where $\lVert \cdot \rVert _{\infty }$ is the supremum norm and $(S_{n}\rho _{\epsilon })(x)=\sum _{i=0}^{n-1}\rho _{\epsilon }(\sigma ^{i}(x))$ for each $x\in X$ . A sequence $\mathcal {F}= \{ \log f_{n} \}_{n=1}^{\infty }$ is subadditive if ${\mathcal F}$ satisfies $f_{n+m}(x) \leq f_{n}(x) f_{m}(\sigma ^{n}_{X} x)$ . The thermodynamic formalism for subadditive sequences was studied by Cao, Feng and Huang [6].

We assume certain regularity conditions on sequences. A sequence $\mathcal {F}= \{ \log f_{n} \}_{n=1}^{\infty }$ has bounded variation if there exists $M \in {\mathbb R}^{+}$ such that $\sup \{ M_{n} : n \in {\mathbb N}\} \leq M$ where

(5) $$ \begin{align} M_{n}= \sup \bigg\{ \frac{f_{n}(x)}{f_{n}(y)} : x,y \in X, x_{i}=y_{i} \textrm{ for } 1 \leq i \leq n\bigg\}. \end{align} $$

More generally, if $\lim _{n \rightarrow \infty }(1/n)\log M_{n}=0$ , then we say that $\mathcal {F}$ has tempered variation. Without loss of generality, we assume $M_{n}\leq M_{n+1}$ for all $n\in {\mathbb N}$ .

A function $f\in C(X)$ belongs to the Bowen class if the sequence ${\mathcal F}$ formed by setting $f_{n}= e^{S_{n}(f)}$ has bounded variation [29]. A function of summable variation belongs to the Bowen class. In this paper, we consider the sequences ${\mathcal F}$ satisfying the following properties.

1. (C1) The sequence ${\mathcal F}^{\kern2pt\prime }:=\{\log (f_{n}e^{C})\}_{n=1}^{\infty }$ is subadditive for some $C\geq 0$ .

2. (C2) There exist $p\in {\mathbb N}$ and $D>0$ such that, given any $u \in B_{n}(X), v \in B_{m}(X)$ , $n, m\in {\mathbb N}$ , there exist $0\leq k\leq p$ and $w \in B_{k}(X)$ such that $uwv\in B_{n+m+k}(X)$ and

   $$ \begin{align*} \sup \{f_{n+m+k}(x):x \in [uwv]\} \geq D \sup \{f_{n}(x):x\in [u]\} \sup \{f_{m}(x):x \in [v]\}. \end{align*} $$

More generally, we have the following property.

1. (D2) There exist $p\in {\mathbb N}$ and a positive sequence $\{D_{n,m}\}_{(n,m)\in {\mathbb N} \times {\mathbb N}}$ such that, given any $u \in B_{n}(X), v \in B_{m}(X)$ , $n, m\in {\mathbb N}$ , there exist $0\leq k\leq p$ and $w \in B_{k}(X)$ such that $uwv\in B_{n+m+k}(X)$ and

   $$ \begin{align*}\sup \{f_{n+m+k}(x):x \in [uwv]\} \geq D_{n,m} \sup \{f_{n}(x):x\in [u]\} \sup \{f_{m}(x):x \in [v]\},\end{align*} $$
   where $\lim _{n\rightarrow \infty }(1/n)\log D_{n,m}=\lim _{m\rightarrow \infty }(1/m)\log D_{n,m}=0$ . Without loss of generality, we assume that $D_{n,m}\geq D_{n,m+1}$ and $D_{n,m}\geq D_{n+1,m}$ .

Remark 2.1. A sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ satisfying (C1) is not always asymptotically additive (see §7). Condition (C2) was introduced by Feng [11] where the thermodynamic formalism of products of matrices was studied. The sequences satisfying (C1) and (C2) with bounded variation generalize almost additive sequences with bounded variation on subshifts with the weak specification property and have been applied to solve questions concerning the Hausdorff dimensions of non-conformal repellers [12, 31]. See [14, 15] for the non-compact case. We will study the sequences satisfying (C1) and (D2) in §§6 and 7.

Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a subadditive sequence of continuous functions on X. For each $n\in {\mathbb N}$ , define

$$ \begin{align*}P_{n}({\mathcal F},\delta)=\sup_{E}\bigg\{\sum_{x\in E}f_{n}(x): E \text { is an } (n, \delta) \text{ separated subset of } X\bigg\}.\end{align*} $$

The topological pressure for ${\mathcal F}$ is defined by

(6) $$ \begin{align} P({\mathcal F}\kern2.5pt) =\lim_{\delta\rightarrow 0}\limsup_{n\rightarrow\infty} \frac{1}{n}\log P_{n}({\mathcal F},\delta). \end{align} $$

Theorem 2.2. [6]

Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a subadditive sequence on X. Then

(7) $$ \begin{align} P({\mathcal F}\kern2.5pt) =\sup_{\mu\in M(X,\sigma_{X})}\bigg\{h_{\mu}(X)+\lim_{n\rightarrow\infty}\frac{1}{n}\int\log f_{n}\,d\kern-1pt\mu\bigg\}. \end{align} $$

A measure $m\in M(X, \sigma _{X})$ is an equilibrium state for ${\mathcal F}$ if the supremum in (7) is attained at m.

Definition 2.3. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a subadditive sequence on X satisfying $P({\mathcal F}\kern2.5pt)\neq -\infty $ . A measure $\mu \in M(X, \sigma _{X})$ is a weak Gibbs measure for ${\mathcal F}$ if there exists $C_{n}>0$ such that

$$ \begin{align*} \frac{1}{C_{n}}< \frac{\mu[x_{1}\ldots x_{n}]}{e^{-nP({\mathcal F}\kern2.5pt)}f_{n}(x)}<C_{n} \end{align*} $$

where $\lim _{n\rightarrow \infty }(1/n)\log C_{n}=0$ , for every $x\in X$ and $n\in {\mathbb N}$ . If there exists $C>0$ such that $C=C_{n}$ for all $n\in {\mathbb N}$ , then $\mu $ is a Gibbs measure.

If $\mu $ is an invariant weak Gibbs measure for a subadditive sequence ${\mathcal F}$ , then it is an equilibrium state for ${\mathcal F}$ . The result of Feng [12, Theorem 5.5] implies the uniqueness of equilibrium states for a class of sequences satisfying (C1) and (C2).

Theorem 2.4. [12]

Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) and (C2) with bounded variation. Then there is a unique invariant Gibbs measure for ${\mathcal F}$ and it is the unique equilibrium state for ${\mathcal F}$ .

Cuneo [9] showed that finding equilibrium states for asymptotically additive sequences is equivalent to that for continuous functions.

Theorem 2.5. (Special case of [9, Theorem 1.2])

Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be an asymptotically additive sequence on X. Then there exists $f\in C(X)$ such that

(8) $$ \begin{align} \lim_{n\rightarrow \infty} \frac{1}{n} \lVert \log f_{n}-S_{n}f \rVert_{\infty}=0. \end{align} $$

Hence, if ${\mathcal F}$ is asymptotically additive, then there exists $f\in C(X)$ such that $\lim _{n\rightarrow \infty } (1/n) {\int}{\log} f_{n} \,d\kern-1pt\mu ={\int}{\kern-1ptf} \,d\kern-1pt\mu $ for every $\mu \in M(X,\sigma _{X}).$ It is clear that (8) implies that ${\mathcal F}$ is asymptotically additive.

3 Subadditive sequences

In this section, we consider Question 1 from §1. Proposition 3.1 is valid for the case when X is a compact metric space and $T: X\rightarrow X$ is a continuous transformation of X. Proposition 3.1 will be applied in the next sections.

Proposition 3.1. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ a subadditive sequence on X. For $h\in C(X)$ , the following conditions are equivalent.

1. (i)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\int \log f_{n} \,d\kern-1pt\mu=\int h\,d\kern-1pt\mu\end{align*} $$
   for every $\mu \in M(X, \sigma _{X}).$

2. (ii)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\int \log f_{n} \,d\kern-1pt\mu=\int h\,d\kern-1pt\mu\end{align*} $$
   for every $\mu \in \mathrm{Erg}(X, \sigma _{X}).$

3. (iii)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0\end{align*} $$
   $\mu $ -almost everywhere on X, for every $\mu \in \mathrm{Erg}(X, \sigma _{X})$ .

Remark 3.2. Proposition 3.1 holds for a sequence ${\mathcal F}$ satisfying (C1) because $\{\log (e^{C}f_{n})\}_{n=1}^{\infty }$ is a subadditive sequence.

Proof. It is clear that (i) implies (ii). By the ergodic decomposition (see [13, Proposition A.1(c)]), (ii) implies (i). Now we assume that (ii) holds. For a measure $\mu \in \mathrm{Erg}(X,\sigma _{X})$ , we obtain

$$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\int \log \bigg(\frac{f_{n}}{e^{(S_{n}h)}}\bigg) \,d\kern-1pt\mu=0. \end{align*} $$

To see that this implies (iii), define $r_{n}(x):=f_{n}(x)/e^{(S_{n}h)(x)}$ . Then $\log r_{n}\in L_{1}(\mu )$ and $\{\log r_{n}\}_{n=1}^{\infty }$ is a subadditive sequence of continuous functions on X. Since $\mu $ is an ergodic measure, by Kingman’s subadditive ergodic theorem, we obtain that $\lim _{n\rightarrow \infty }(1/n)\log r_{n}(x)=\lim _{n\rightarrow \infty }(1/n)\int \log r_{n} \,d\kern-1pt\mu =0 \mu $ -almost everywhere on X. Now we assume that (iii) holds. Given $\mu \in \mathrm{Erg}(X,\sigma _{X})$ , applying the subadditive ergodic theorem to the sequence $\{\log r_{n}\}_{n=1}^{\infty }$ , we obtain

$$ \begin{align*} \begin{split} \int \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}}{e^{(S_{n}h)}} \bigg) \,d\kern-1pt\mu&=\lim_{n\rightarrow\infty} \frac{1}{n}\int \log \bigg( \frac{f_{n}}{e^{(S_{n}h)}} \bigg)\,d\kern-1pt\mu \\ &=\lim_{n\rightarrow\infty} \bigg(\frac{1}{n}\int \log {f_{n}}\,d\kern-1pt\mu- \int h \,d\kern-1pt\mu\bigg). \end{split} \end{align*} $$

Hence, we obtain (ii).

4 Subadditive sequences which are asymptotically additive

Subadditive sequences are not always asymptotically additive. In this section we study a class of subadditive sequences on shift spaces (compact spaces) which are also asymptotically additive. The goal of this section is to characterize such sequences using a particular property for periodic points. The results in this section are applied in §§6 and 7 to study relative pressure functions.

Lemma 4.1. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) with tempered variation. Suppose that ${\mathcal F}$ satisfies the following two conditions (i) and (ii).

1. (i) There exists $h\in C(X)$ such that

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0\end{align*} $$
   for every periodic point $x\in X$ .

2. (ii) There exist $k, N\in {\mathbb N}$ and a sequence $\{M_{n}\}_{n=1}^{\infty }$ of positive real numbers satisfying $ \lim _{n\rightarrow \infty }({1}/{n})\log M_{n}=0$ such that, for given any $u\in B_{n}(X), n \geq N$ , there exist $0 \leq q \leq k$ and $w\in B_{q}(X)$ such that $z:=(uw)^{\infty }$ is a point in X satisfying

   (9) $$ \begin{align} f_{j(n+q)}(z)\geq (M_{n} \sup\{f_{n}(x): x\in [u]\})^{j} \end{align} $$
   for every $j\in {\mathbb N}$ .

Then ${\mathcal F}$ is an asymptotically additive sequence on X.

Remark 4.2. Let $(X,\sigma _{X})$ be an irreducible shift of finite type and k be a weak specification number. Then for each $u\in B_{n}(X)$ there exist $0\leq q\leq k$ and $w\in B_{q}(X)$ such that $(uw)^{\infty }\in X$ .

Proof. Suppose that (i) and (ii) hold. We will show that

(10) $$ \begin{align} \lim_{n\rightarrow\infty}\frac{1}{n} \bigg\lVert \log \bigg(\frac{f_{n}}{e^{(S_{n}h)}}\bigg)\bigg\rVert {}_{\infty}=0. \end{align} $$

Let $k, M_{n}, N$ be defined as in (ii). For $h\in C(X)$ , let

(11) $$ \begin{align} M^{h}_{n}:=\sup\bigg\{\frac{e^{(S_{n}h)(x)}}{e^{(S_{n}h)(x^{\prime})}}: x_{i}=x^{\prime}_{i}, 1\leq i\leq n\bigg\} \end{align} $$

for each $n\in {\mathbb N}$ and $C_{h}:=\max _{0\leq i \leq k}\{(S_{i}h)(x): x\in X\}$ , where $(S_{0}h)(x):=1$ for every $x\in X$ . Let $\epsilon>0$ . Take $N_{1}\in {\mathbb N}$ large enough so that

$$ \begin{align*}\frac{1}{n}\vert \log ({M^{h}_{n}}{e^{C_{h}}})\vert<\epsilon,\quad \frac{1}{n}\vert \log M_{n}\vert<\epsilon \quad\text{and}\quad \frac{n}{n+k}>\frac{1}{2} \end{align*} $$

for all $n>N_{1}$ . Let $N_{2}=\max \{N, N_{1}\}$ and let $n\geq N_{2}$ . Then, for $x_{1}\ldots x_{n}\in B_{n}(X)$ , there exists $w\in B_{q}(X), 0\leq q\leq k,$ such that $y^{*}:=(x_{1},\ldots , x_{n}, w)^{\infty }\in X$ satisfying (9). Since $y^{*}$ is a periodic point, (i) implies that there exists $N(y^{*}) \in {\mathbb N}$ such that

$$ \begin{align*} \frac{1}{i}\bigg\vert \log \bigg(\frac{f_{i}(y^{*})}{e^{(S_{i}h)(y^{*})}}\bigg)\bigg\vert< \epsilon\end{align*} $$

for all $i> N(y^{*})$ . Take $j> N(y^{*})$ . By (ii), for $ z \in [x_{1}\ldots x_{n}]$ , we obtain

$$ \begin{align*} \epsilon&>\frac{1}{j(n+q)} \log \bigg(\frac{f_{j(n+q)}(y^{*})}{e^{(S_{j(n+q)}h)(y^{*})}}\bigg) \geq \frac{1}{j(n+q)}\log \bigg(\frac{M_{n}f_{n}(z)}{M^{h}_{n}e^{(S_{n}h)(z)}e^{C_{h}}}\bigg)^{j}\\ &=\frac{1}{(n+q)}\log M_{n} +\frac{1}{(n+q)}\log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg) -\frac{1}{(n+q)}\log ({M^{h}_{n}}{e^{C_{h}}})\\ &>-2\epsilon +\frac{1}{n+q} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)>-2\epsilon +\frac{n}{n+q} \bigg (\frac{1}{n} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)\bigg). \end{align*} $$

Without loss of generality assume $(1/n)\log (f_{n}(z)/e^{(S_{n}h)(z)})>0$ . Hence, for any $x_{1}\ldots x_{n}\in B_{n}(X), n\geq N_{2}, z\in [x_{1}\ldots x_{n}]$ , we obtain that $({1}/{n}) \log ({f_{n}(z)}/ e^{(S_{n}h)(z)}) <6\epsilon .$

Next we show that there exists $N^{\prime }\in {\mathbb N}$ such that, for all $z\in [x_{1},\ldots , x_{n}], n\geq N^{\prime }$ , $({1}/{n}) \log ({f_{n}(z)}/e^{(S_{n}h)(z)})>-4\epsilon .$ Since ${\mathcal F}$ has tempered variation, for each $n\in {\mathbb N}$ , let $M^{{\mathcal F}}_{n}:=\sup \{f_{n}(x)/f_{n}(x^{\prime }): x_{i}=x^{\prime }_{i}, 1\leq i\leq n\}$ . Let $C_{{\mathcal F}}:=\max _{0\leq i\leq k}\{f_{i}(x): x\in X\}$ , where $f_{0}(x):=1$ for every $x\in X$ , and $\bar {C}_{h}:=\min _{0\leq i\leq k}\{(S_{i}h)(x): x\in X\}$ . Let C be defined as in (C1). Take $N_{3}\in {\mathbb N}$ large enough so that

$$ \begin{align*}\frac{1}{n}\vert \log ({M^{{\mathcal F}}_{n}}{M^{h}_{n}}C_{{\mathcal F}}{e^{-{\bar{C}}_{h}+2C}})\vert<\epsilon \quad\text{and}\quad \frac{n}{n+k}>\frac{1}{2}\end{align*} $$

for all $n>N_{3}$ . Since ${\mathcal F}$ satisfies (C1), we obtain that

$$ \begin{align*} \begin{split} \frac{f_{j(n+q)}(y^{*})}{e^{(S_{j(n+q)}h)(y^{*})}} &\leq \bigg(\frac{C_{{\mathcal F}}M^{h}_{n}e^{2C}\sup\{f_{n}(y):y\in [x_{1}\ldots x_{n}]\}} {e^{\bar{C}_{h}}\sup\{e^{(S_{n}h)(y)}: y\in [x_{1}\ldots x_{n}]\}}\bigg)^{j}\\ &\leq \bigg(\frac{C_{{\mathcal F}}M^{h}_{n}M^{{\mathcal F}}_{n}e^{2C}f_{n}(z)} {e^{\bar{C}_{h}+(S_{n}h)(z)}}\bigg)^{j}, \end{split} \end{align*} $$

where in the last inequality z is a point from the cylinder set $[x_{1}\ldots x_{n}]$ . Hence, for $j> N(y^{*})$ ,

$$ \begin{align*} \begin{split} -\epsilon&<\frac{1}{j(n+q)} \log \bigg(\frac{f_{j(n+q)}(y^{*})}{e^{(S_{j(n+q)}h)(y^{*})}}\bigg)\\ &<\frac{1}{n+q} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg) +\frac{1}{n+q} \log ({M^{{\mathcal F}}_{n}}{M^{h}_{n}}C_{{\mathcal F}}{e^{-{\bar{C}}_{h}+2C}})\\ &<\frac{1}{n+q} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)+\epsilon\\ &=\frac{n}{n+q} \bigg(\frac{1}{n} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)\bigg)+\epsilon. \end{split} \end{align*} $$

Without loss of generality assume $(1/n)\log (f_{n}(z)/e^{(S_{n}h)(z)})<0$ . For all $z \in [x_{1}\ldots x_{n}], n \geq N_{3}$ , we obtain that $(1/n)\log (f_{n}(z)/e^{(S_{n}h)(z)})>-4\epsilon .$ Hence, we obtain (10).

By Lemma 4.1, we obtain some conditions for a sequence ${\mathcal F}$ satisfying (C1) to be asymptotically additive, assuming that Lemma 4.1 (ii) is satisfied.

Theorem 4.3. Let $(X,\sigma _{X})$ be a subshift. Let ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) with tempered variation and Lemma 4.1 (ii). Then the following statements are equivalent for $h\in C(X)$ .

1. (i) ${\mathcal F}$ is asymptotically additive on X satisfying

   $$ \begin{align*}\lim_{n\rightarrow \infty} \frac{1}{n} \bigg\lVert \log \bigg (\frac{f_{n}}{e^{(S_{n} h)}}\bigg) \bigg\rVert_{\infty}=0.\end{align*} $$

2. (ii)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\int \log f_{n} \,d\kern-1pt\mu=\int h\,d\kern-1pt\mu\end{align*} $$
   for every $\mu \in M(X, \sigma _{X}).$

3. (iii)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0\end{align*} $$
   for every periodic point $x\in X$ .

4. (iv)

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0\end{align*} $$
   for every $x\in X$ .

Proof. The implications (i) $\Longrightarrow $ (ii) $\Longrightarrow $ (iii) are clear by applying Theorem 2.5 and Proposition 3.1. To see (iii) $\Longrightarrow $ (iv) $\Longrightarrow $ (i) , we apply Lemma 4.1.

In the next theorem we study an equivalent condition for a subadditive sequence ${\mathcal F}$ to be an asymptotically additive sequence.

Theorem 4.4. Let $(X,\sigma _{X})$ be an irreducible shift of finite type and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) with tempered variation. Then ${\mathcal F}$ is asymptotically additive on X if and only if the following two conditions hold.

1. (i) There exists $h\in C(X)$ such that

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0\end{align*} $$
   for every periodic point $x\in X$ .

2. (ii) There exist $k\in {\mathbb N}, c\geq 0$ and a sequence $\{M_{n}\}_{n=1}^{\infty }$ of positive real numbers satisfying $\lim _{n\rightarrow \infty }(1/{n})\log M_{n}=0$ such that the following property, which we refer to as property (P), holds. For every $0<\epsilon <1$ , there exists $N\in {\mathbb N}$ such that, given any $u\in B_{n}(X), n\geq N$ , there exist $0\leq q\leq k$ and $w\in B_{q}(X)$ such that $z:=(uw)^{\infty }$ is a point in X satisfying

   (12) $$ \begin{align} f_{j(n+q)}(z)\geq (M_{n}e^{-cn\epsilon})^{j} (\sup\{f_{n}(x): x\in [u]\})^{j} \end{align} $$
   for every $j\in {\mathbb N}$ .

Theorem 4.3 holds if we replace Lemma 4.1 (ii) by condition (ii) above.

Remark 4.5. Theorem 4.4 (ii) is a generalization of Lemma 4.1 (ii). If we set $c=0$ in (12), we obtain (9).

Proof. Assume that ${\mathcal F}$ is asymptotically additive. Then (i) is obvious and for a given $0<\epsilon <1$ there exists $N\in {\mathbb N}$ such that for all $n\geq N$ ,

(13) $$ \begin{align} e^{-n\epsilon+(S_{n}h)(x)}<f_{n}(x)<e^{n\epsilon+(S_{n}h)(x)} \end{align} $$

for all $x\in X$ . Since $(X,\sigma _{X})$ is an irreducible shift of finite type, let k be a weak specification number. Then for $x_{1}\ldots x_{n}\in B_{n}(X), n\geq N$ , there exists $w\in B_{q}(X), 0\leq q\leq k$ , such that $y^{*}:=(x_{1}, \ldots , x_{n}, w)^{\infty }\in X$ . Let $ \bar {C}_{h}, M^{h}_{n}$ and $M^{{\mathcal F}}_{n}$ be defined as in the proof of Lemma 4.1. Then for any $z\in [x_{1}\ldots x_{n}], j\in {\mathbb N}$ ,

$$ \begin{align*} \begin{split} f_{(n+q)j}(y^{*})&\geq e^{-j(n+q)\epsilon+(S_{j(n+q)}h)(y^{*})} \geq e^{-j(n+q)\epsilon}\cdot\bigg(\frac{1}{M^{h}_{n}}e^{(S_{n}h)(z)}e^{\bar{C}_{h}}\bigg)^{j}\\ &\geq \bigg(\frac{1}{M^{h}_{n}}e^{-2\epsilon n-k\epsilon+\bar{C}_{h}}\bigg)^{j} f_{n}^{j}(z) \geq \bigg(\frac{1}{M^{h}_{n}}e^{-2\epsilon n-k+\bar{C}_{h}}\bigg)^{j} f_{n}^{j}(z). \end{split} \end{align*} $$

Setting $c=2$ and $M_{n}= e^{-k+\bar {C}_{h}}/(M^{{\mathcal F}}_{n}M^{h}_{n})$ , we obtain (ii). Now we show the reverse implication. We slightly modify the proof of Lemma 4.1 by taking account of property (P). We only consider the case when $c>0$ . Let $C_{h}$ and $M^{h}_{n}$ be defined as in the proof of Lemma 4.1. Let $0<\epsilon <1$ be fixed. By (ii), there exists $N^{\prime }\in {\mathbb N}$ such that

$$ \begin{align*} -\frac{3c}{2}\epsilon<\frac{1}{n+i}\log (e^{-nc\epsilon}M_{n})<-\frac{c}{2}\epsilon,\quad \frac{1}{n}\vert \log ({M^{h}_{n}}{e^{C_{h}}})\vert<\epsilon \quad\text{and}\quad \frac{n}{n+k}>\frac{1}{2}\end{align*} $$

for all $n>N^{\prime }, 0\leq i\leq k$ . In the proof of Lemma 4.1, define $N_{2}:=\max \{N, N^{\prime }\}$ . Replacing $M_{n}$ by $e^{-nc\epsilon }M_{n}$ in the proof of Lemma 4.1, we obtain that for any $x_{1}\ldots x_{n}\in B_{n}(X), n\geq N_{2}, z\in [x_{1}\ldots x_{n}]$ ,

$$ \begin{align*} \frac{1}{n} \log \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)<(4+3c)\epsilon.\end{align*} $$

Using the latter part of the proof of Lemma 4.1, we obtain the results.

5 Asymptotically additive sequences and subadditive sequences satisfying (C1) and (C2)

In this section, we study the sequences ${\mathcal F}$ on subshifts X with bounded variation satisfying (C1) and (C2). Since there exists a unique Gibbs equilibrium state m for such a sequence ${\mathcal F}$ (Theorem 2.4), we study the condition for m to be an invariant Gibbs measure for some continuous function. In Theorem 5.6, we also characterize the form of sequences ${\mathcal F}$ in terms of the properties of equilibrium states.

Theorem 5.1. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) and (C2) with bounded variation. Let m be the unique invariant Gibbs measure for ${\mathcal F}$ . Then the following statements are equivalent.

1. (i) There exists $h\in C(X)$ such that

   $$ \begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0 \end{align*} $$
   for every $x\in X$ .

2. (ii) ${\mathcal F}$ is asymptotically additive on X.

3. (iii) The measure m is an invariant weak Gibbs measure for a continuous function on X.

Remark 5.2.

1. (1) There exists a sequence ${\mathcal F}$ which satisfies (C1), (C2) with bounded variation satisfying Theorem 5.1 (ii). On the other hand, there exists a sequence ${\mathcal F}$ with bounded variation satisfying (C1) and (C2) without being asymptotically additive (see §7).

2. (2) If $h\in C(X)$ in (i) exists, then m is a unique equilibrium state for h.

To prove Theorem 5.1, we apply the following lemmas. We continue to use ${\mathcal F}$ and m defined as in Theorem 5.1. In the next lemma we first study the relation between Theorem 5.1 (i) and (ii).

Lemma 5.3. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) and (C2) with bounded variation. If there exists $h\in C(X)$ such that

(14) $$ \begin{align} \lim_{n\rightarrow\infty}\frac{1}{n}\log \bigg(\frac{f_{n}(x)}{e^{(S_{n}h)(x)}}\bigg)=0 \end{align} $$

for every $x \in X$ , then ${\mathcal F}$ is asymptotically additive on X.

Remark 5.4. Lemma 5.3 implies that if ${\mathcal F}$ satisfies the assumptions of the lemma then uniform convergence of the sequence of functions $\{{1}/{n}\log ({f_{n}}/{e^{(S_{n}h)}})\}_{n=1}^{\infty }$ is equivalent to pointwise convergence of the sequence of functions.

Proof. Let $\epsilon>0$ . It is enough to show that there exists $N\in {\mathbb N}$ such that for any $z~\in ~[u], u\in B_{n}(X), n>N$ ,

$$ \begin{align*} -\epsilon<\frac {1}{n}\log \bigg (\frac{f_{n} (z)}{e^{(S_{n} h) (z)}}\bigg)<\epsilon. \end{align*} $$

Let p be defined as in (C2). Let $\overline {m}_{h}:=\max _{0\leq l \leq p}\{e^{(S_{l}h)(x)}: x\in X\}$ , where $(S_{0}h)(x):=1$ for every $x\in X$ . Since h has tempered variation, let $M^{h}_{n}$ be defined as in (11). Let M be a constant defined as in the definition of bounded variation and D be defined as in (C2). Then there exists $N_{1}\in {\mathbb N}$ such that

(15) $$ \begin{align} \frac{1}{n} \log M <\epsilon, \!\quad\frac{1}{n} \log M^{h}_{n} <\epsilon, \!\quad\frac{1}{n}\bigg\vert \log \frac{1}{\overline{m}_{h}}\bigg\vert<\epsilon, \!\quad\frac{1}{n}\vert \log D\vert<\epsilon \quad\text{and}\quad \frac{n}{n+p}>\frac{1}{2}, \end{align} $$

for all $n>N_{1}$ . Take $n>N_{1}$ . Condition (C2) implies that for a given $u \in B_{n}(X)$ , there exists $w_{1}\in B_{l_{1}}(X), 0\leq l_{1}\leq p$ such that for any $x\in [uw_{1}u], z\in [u]$ ,

$$ \begin{align*} \sup\{f_{2n+l_{1}}(x): x\in [uw_{1}u]\} \geq D (\sup\{f_{n}(x): x\in [u]\})^{2}\geq D f^{2}_{n}(z). \end{align*} $$

Repeating this, given $j\geq 2, u \in B_{n}(X)$ , there exist allowable words $w_{i}$ of length $l_{i}$ , $1\leq i\leq j-1, 0\leq l_{i}\leq p,$ such that $uw_{1}uw_{2}u\ldots uw_{j-1}u$ is an allowable word of length $jn+\sum _{i=1}^{j-1}l_{i}$ satisfying that, for any $x\in [uw_{1}uw_{2}u\ldots uw_{j-1}u]$ and $z\in [u]$ ,

(16) $$ \begin{align} Mf_{jn+\sum_{i=1}^{j-1}l_{i}}(x)\geq \sup\{f_{jn+\sum_{i=1}^{j-1}l_{i}}(x): x\in [uw_{1}uw_{2}u\ldots uw_{j-1}u]\} \geq D^{j-1} {f_{n}(z)}^{j}. \end{align} $$

By the additivity of the sequence $\{{S_{n}h}\}_{n=1}^{\infty }$ ,

(17) $$ \begin{align} e^{(S_{jn+\sum_{i=1}^{j-1}l_{i}}h)(x)}\leq (M^{h}_{n}e^{S_{n}h(z)})^{j}{\overline{m}_{h}}^{j-1}. \end{align} $$

Hence, by (16) and (17) we obtain for $j\geq 2$ , $x\in [uw_{1}uw_{2}u\ldots uw_{j-1}u]$ and $z\in [u]$ ,

(18) $$ \begin{align} \frac{f_{jn+\sum_{i=1}^{j-1}l_{i}}(x)}{e^{(S_{jn+\sum_{i=1}^{j-1}l_{i}}h)(x)}}\geq \bigg(\frac{1}{M^{h}_{n}}\bigg)^{j} \bigg(\frac{f_{n}(z)}{e^{(S_{n}h)(z)}}\bigg)^{j} \bigg(\frac{D}{\overline{m}_{h}}\bigg)^{j-1}\cdot \frac{1}{M}. \end{align} $$

Let $c_{1}=[u w_{1}u],\ldots , c_{i}=[uw_{1} uw_{2}u\ldots u w_{i} u], i\in {\mathbb N}$ . Then by Cantor’s intersection theorem $\bigcap _{i\in {\mathbb N}} c_{i}\neq \emptyset $ and it consists of exactly one point in X. We call it $x^{*}\in X$ . For each $y\in X$ , define $A_{n}(y):=f_{n}(y)/e^{(S_{n}h)(y)}$ . By assumption (14), there exists $t(x^{*})\in {\mathbb N}$ , which depends on $x^{*}$ such that for all $i\geq t(x^{*})$ ,

$$ \begin{align*} -\epsilon<\frac {1}{i}\log A_{i}(x^{*})<\epsilon. \end{align*} $$

Letting $s(u, j):=\sum _{i=1}^{j-1}l_{i}$ , for $j\geq t(x^{*})\geq 2$ , and using (15) and (18), we obtain

$$ \begin{align*} \begin{split} \epsilon&>\frac{1}{jn+s(u, j)} \log A_{jn+s(u, j)} (x^{*}) \\ &\geq \frac{1}{n+({1}/{j})s(u,j)}\log \frac{1}{M^{h}_{n}}+\frac{1-{1}/{j}}{n+({1}/{j})s(u,j)} \log \frac{1}{\overline{m}_{h}}+ \frac{1}{jn+s(u, j)}\log \frac{1}{M} \\ &\quad + \frac{n(j-1)}{jn+s(u,j)}\cdot \frac{1}{n} \log D+\frac{n}{n+({1}/{j})s(u,j)}\cdot\frac{1}{n}\log A_{n}(z).\\ \end{split} \end{align*} $$

Without loss of generality, assume $\log A_{n}(z)> 0.$ By a simple calculation, we obtain that

(19) $$ \begin{align} \frac{1}{n}\log A_{n}(z)<10\epsilon \end{align} $$

for all $n>N_{1}, z\in [u]$ , for any $u\in B_{n}(X)$ .

Next we will show that there exists $N_{2}\in {\mathbb N}$ such that

(20) $$ \begin{align} -6\epsilon < \frac{1}{n}\log A_{n}(z) \end{align} $$

for all $n>N_{2}, z\in [u]$ for any $u\in B_{n}(X)$ . Define $f_{0}(x):=1$ . Let $\overline {M}:=\max _{0\leq i\leq p}\{ f_{i}(x):x\in X\}$ and $\overline {m}_{1}:= \min _{0\leq k\leq p} \{e^{(S_{k} h) (x)}: x \in X\}.$ Take $N_{2}$ so that

(21) $$ \begin{align} \frac{1}{n}\vert \log (MM^{h}_{n})\vert<\epsilon, \frac{1}{n}\bigg\vert \log \bigg(\frac{\overline{M} e^{2C}}{\overline{m}_{1}}\bigg) \bigg\vert<\epsilon, \frac{n}{n+p}>\frac{1}{2} \end{align} $$

for all $n>N_{2}$ . For $n>N_{2}$ , let $u\in B_{n}(X)$ . Construct $x\in [uw_{1}uw_{2}\ldots uw_{j-1}u], j\geq 2,$ as in the above argument and let $z\in [u]$ . Using (C1), it is easy to obtain for each $j\geq 2$ ,

(22) $$ \begin{align} f_{jn+\sum_{i=1}^{j-1}l_{i}}(x)\leq (\overline{M}e^{2C})^{j-1}(Mf_{n}(z))^{j} \end{align} $$

and

(23) $$ \begin{align} e^{(S_{jn+\sum_{i=1}^{j-1}l_{i}}h)(x)}\geq \bigg(\frac{e^{(S_{n}h)(z)}}{M^{h}_{n}}\bigg)^{j}(\overline{m}_{1})^{j-1}. \end{align} $$

Define $x^{*}\in X$ as before. For all $j\geq t(x^{*})$ , by using (21), (22) and (23), we obtain

$$ \begin{align*} -\epsilon< \frac{1}{jn+s(u, j)} \log A_{jn+s(u, j)} (x^{*}) < 2\epsilon +\frac{n}{n+({1}/{j})s(u,j)}\cdot \frac{1}{n}\log A_{n}(z). \end{align*} $$

Without loss of generality, assuming that $\log A_{n}(z)<0$ , we obtain (20) for all $n>N_{2}$ , each $z\in [u], u\in B_{n}(X)$ . The result follows by (19) and (20).

Lemma 5.5. Under the assumptions of Theorem 5.1, ${\mathcal F}$ is asymptotically additive if and only if there exists a continuous function for which m is an invariant weak Gibbs measure.

Proof. Suppose ${\mathcal F}$ is asymptotically additive. Then by [9, Theorem 1.2] there exist $h, u_{n}\in C(X)$ , $n\in {\mathbb N}$ , such that $f_{n}(x)=e^{(S_{n}h)(x)+u_{n}(x)}$ satisfying $\lim _{n\rightarrow \infty }({1}/{n})\vert \vert u_{n} \vert \vert _{\infty } =0$ . Since there exists a constant $C>0$ such that

(24) $$ \begin{align} \frac{1}{C} \leq \frac{m [x_{1}\ldots x_{n}]}{e^{-nP({\mathcal F}\kern2.5pt)}f_{n}(x)} \leq {C} \end{align} $$

for each $x\in [x_{1}\ldots x_{n}]$ , replacing $f_{n} (x)$ by $e^{(S_{n}h)(x)+u_{n}(x)}$ , we obtain

$$ \begin{align*} \frac{1}{Ce^{\vert \vert u_{n} \vert \vert {}_{\infty}}}\leq \frac{e^{u_{n}(x)}}{C} \leq \frac{m [x_{1}\ldots x_{n}]}{e^{-nP({\mathcal F}\kern2.5pt)+(S_{n}h)(x)}} \leq Ce^{u_{n}(x)}\leq Ce^{\vert \vert u_{n} \vert \vert {}_{\infty}}. \end{align*} $$

Set $A_{n}=Ce^{\vert \vert u_{n} \vert \vert _{\infty }}$ . Since $ \lim _{n\rightarrow \infty }(1/n)\int \log f_{n} \,d\kern-1pt\mu =\int h\,d\kern-1pt\mu $ for every $\mu \in M(X, \sigma _{X})$ , we obtain that $P({\mathcal F}\kern2.5pt)=P(h)$ . Conversely, assume that m is an invariant weak Gibbs measure for $\tilde h\in C(X)$ . Hence, there exists $C_{n}>0$ such that

(25) $$ \begin{align} \frac{1}{C_{n}} \leq \frac{m [x_{1}\ldots x_{n}]}{e^{-nP(\tilde{h})+(S_{n}\tilde{h})(x)}}\leq {C_{n}} \end{align} $$

for all $x \in [x_{1}\ldots x_{n}]$ , where $\lim _{n\rightarrow \infty }(1/n)\log C_{n}=0$ . Since m is the Gibbs measure for  ${\mathcal F}$ ,

(26) $$ \begin{align} \frac{1}{C} \leq \frac{m [x_{1}\ldots x_{n}]}{e^{-nP({\mathcal F}\kern2.5pt)}f_{n}(x)} \leq C \end{align} $$

for some $C>0$ . Using (25) and (26), we obtain

(27) $$ \begin{align} \frac{1}{C_{n}C}\leq \frac{f_{n}(x)}{e^{(S_{n}({\tilde h-P(\tilde h)+P({\mathcal F}\kern2.5pt))})(x)}}\leq C_{n}C, \end{align} $$

for all $x \in [x_{1}\ldots x_{n}]$ . Hence, by [9, Theorem 1.2], ${\mathcal F}$ is an asymptotically additive sequence.

Proof of Theorem 5.1. By [9, Theorem 1.2], (ii) implies (i). Theorem 5.1 follows by Lemmas 5.3 and 5.5.

Theorem 5.6. Let $(X,\sigma _{X})$ be a subshift and ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ be a sequence on X satisfying (C1) and (C2) with bounded variation. Let m be the unique invariant Gibbs measure for ${\mathcal F}$ . Suppose that one of the equivalent statements in Theorem 5.1 holds. Then the following statements hold.

1. (i) There exits a sequence $\{C_{n, m}\}_{n,m\in {\mathbb N}}$ such that

   (28) $$ \begin{align} \frac{1}{C_{n,m}}\kern1.4pt{\leq}\kern1.4pt \frac{f_{n+m}(x)}{f_{n}(x) f_{m}(\sigma^{n}_{X} x)}\kern1.4pt{\leq}\kern1.4pt C_{n,m}, \!\quad\text{where} \lim_{n\rightarrow \infty}\frac{1}{n}\log C_{n,m}\kern1.4pt{=}\kern1.4pt \lim_{m\rightarrow \infty}\frac{1}{m}\log C_{n,m}\kern1.4pt{=}\kern1.4pt 0. \end{align} $$

2. (ii) If m is a Gibbs measure for a continuous function, then ${\mathcal F}$ is an almost additive sequence on X.

Hence, if there is no sequence $\{C_{n, m}\}_{n,m\in {\mathbb N}}$ satisfying (28), then there exists no continuous function for which m is an invariant weak Gibbs measure.

Remark 5.7.

1. (1) In Example 7.2, we study a sequence which satisfies (C1) and (C2) without (28).

2. (2) See [3, Theorem 1.14(ii)] for the result related to (i). (ii) was also obtained in [9, §4.1] since m satisfies the quasi-Bernoulli property (see [9]).

Proof. Let h be defined as in Theorem 5.1 (i). By the proofs of Lemmas 5.3 and 5.5, we obtain that $P({\mathcal F}\kern2.5pt)=P(h)$ and m is an invariant weak Gibbs measure for h. Replacing $\tilde h$ by h in (27), we obtain that

(29) $$ \begin{align} \frac{1}{C^{3}C_{n}C_{m}C_{n+m}}\leq \frac{f_{n+m}(x)}{f_{n}(x) f_{m}(\sigma^{n}_{X} x)}\leq C^{3}C_{n+m}C_{n}C_{m}. \end{align} $$

Since $\lim _{n\rightarrow \infty } (1/n)\log C_{n}=0$ , by setting $C_{n,m}:=C^{3}C_{n}C_{m}C_{n+m}$ we obtain the first statement. To obtain the second statement, we apply the latter part of the proof of Lemma 5.5. By replacing $C_{n}$ in (25) and (27) by a constant, we obtain the second statement. The last statement follows from Theorem 5.1.

6 Relation between the existence of a continuous compensation function and an asymptotically additive sequence

In this section we consider relative pressure functions $P(\sigma _{X}, \pi , f)$ , where $f\in C(X)$ . In general we can represent $P(\sigma _{X}, \pi , f)$ by using a subadditive sequence satisfying (D2). What are necessary and sufficient conditions for the existence of $h\in C(Y)$ satisfying ${\int}{P}(\sigma _{X}, \pi , f) \,dm={\int}{h} \,dm$ for each $m\in M(Y, \sigma _{Y})$ ? By [9, Theorem 2.1], if $P(\sigma _{X}, \pi , f)$ is represented by an asymptotically additive sequence then we can find such a function h. We will study necessary conditions for the existence of such a function h and relate them with the existence of a compensation function for a factor map between subshifts. To this end, we will apply the results from §4. We will study the property for periodic points from Lemma 4.1(ii).

Theorem 6.1. (Relativized variational principle [17])

Let $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ be subshifts and $\pi :X\rightarrow Y$ be a one-block factor map. Let $f\in C(X)$ . Then for $m\in M(Y, \sigma _{Y})$ ,

(30) $$ \begin{align} \int P(\sigma_{X}, \pi,f) \,dm= \sup\bigg\{h_{\mu}(\sigma_{X})-h_{m}(\sigma_{Y}) +\int f \,d\kern-1pt\mu: \mu\in M(X, \sigma_{X}),\pi\mu=m\bigg\}. \end{align} $$

Applying the relativized variational principle, we first study Borel measurable compensation functions for factor maps between subshifts.

Proposition 6.2. Let $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ be subshifts and $\pi :X\rightarrow Y$ be a one-block factor map. For each $f\in C(X)$ , $f-P(\sigma _{X}, \pi , f)\circ \pi $ is a Borel measurable compensation function for $\pi $ .

Remark 6.3. In general, $f-P(\sigma _{X}, \pi , f)\circ \pi $ is not continuous on X.

Proof. Let $m\in M(Y, \sigma _{Y})$ and $\phi \in C(Y)$ . Applying Theorem 6.1, we obtain

$$ \begin{align*} \begin{split} &\sup\kern-1.2pt\bigg\{\kern-1pt h_{\mu}(\sigma_{X}) \kern1pt{-}\kern-1pt\!\int\!\kern-1.2pt P(\sigma_{X}, \pi, f)\circ\pi \,d\kern-1pt\mu +\!\int\!\kern-1.2pt f \,d\kern-1pt\mu +\! \int\!\kern-1.2pt \phi\circ \pi \,d\kern-1pt\mu: \mu\kern1.3pt{\in}\kern1.3pt M(X, \sigma_{X}), \pi \mu=m\kern-1.2pt\bigg\}\\ &\quad= \sup\kern-1.2pt\bigg\{\kern-1.2pt h_{\mu}(\sigma_{X}) +\int\! f \,d\kern-1pt\mu: \mu\in M(X, \sigma_{X}), \pi \mu=m\kern-1.2pt\bigg\} \kern-1.2pt{-}\kern-1.2pt\int\!\kern-1.2pt P(\sigma_{X}, \pi, f)\,dm\kern1.5pt{+}\! \int\! \phi \,dm\\ &\quad=h_{m}(\sigma_{Y})+\int \phi \,dm. \end{split} \end{align*} $$

Taking the supremum over $m \in M(Y, \sigma _{Y})$ , we obtain

(31) $$ \begin{align} &\sup\bigg\{h_{\mu}(\sigma_{X}) -\int P(\sigma_{X}, \pi, f)\circ\pi \,d\kern-1pt\mu +\int f \,d\kern-1pt\mu + \int \phi\circ \pi \,d\kern-1pt\mu: \mu\in M(X, \sigma_{X})\bigg\}\nonumber\\ &\quad=\sup \bigg\{ h_{m}(\sigma_{Y})+\int \phi \,dm: m \in M(Y, \sigma_{Y})\bigg\}. \end{align} $$

Let $\pi :X\rightarrow Y$ be a one-block factor map between subshifts. For $y=(y_{i})_{i=1}^{\infty }$ , let $E_{n}(y)$ be a set consisting of exactly one point from each cylinder $[x_{1}\ldots x_{n}]$ in X such that $\pi (x_{1}\ldots x_{n})=y_{1}\ldots y_{n}$ . For $n\in {\mathbb N}$ and $f\in C(X)$ , define

(32) $$ \begin{align} g_{n}(y)=\sup_{E_{n}(y)}\bigg\{\sum_{x\in E_{n}(y)}e^{(S_{n}f)(x)}\bigg\}. \end{align} $$

The following result can be deduced by [12, Proposition 3.7(i)]. If $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ are subshifts and $\pi :X\rightarrow Y$ is a one-block factor map, then for $f\in C(X)$ ,

(33) $$ \begin{align} P(\sigma_{X}, \pi, f)(y)=\limsup {}_{n\rightarrow\infty} \frac{1}{n} \log {g_{n}}(y) \end{align} $$

$\mu $ -almost everywhere for every invariant Borel probability measure $\mu $ on Y. Equation (33) was shown by Petersen and Shin [19] for the case when X is an irreducible shift of finite type. The result for general subshifts is obtained by combining [12, Proposition 3.7(i)] and the fact that $P(\sigma _{X}, \pi , f)(y)\leq \limsup _{n\rightarrow \infty } (1/{n}) \log {g_{n}}(y)$ for all $y\in Y$ . Note that the function $P(\sigma _{X}, \pi , f)$ is bounded on Y.

Lemma 6.4. Let $(X, \sigma _{X})$ be a subshift with the weak specification property, $(Y, \sigma _{Y})$ be a subshift and $\pi :X\rightarrow Y$ be a one-block factor map. If $f\in C(X)$ , then the sequence ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ on Y satisfies (C1) and (D2) with bounded variation.

Remark 6.5. In particular, the sequence $\{\log g_{n}\}_{n=1}^{\infty }$ on Y satisfies (C1) and (C2) with bounded variation if $f\in C(X)$ is in the Bowen class (see [12, 14, 32]).

Proof. First we show that ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ satisfies (C1). Let $y=(y_{1},\ldots , y_{n}, \ldots , y_{n+m}, \ldots )\in Y$ . For each $x\in E_{n+m}(y)$ , define $S_{x}$ by $S_{x}:=\{x^{\prime }\in E_{n+m}(y): x^{\prime }_{i}=x_{i} , 1\leq i\leq n\}$ . Take a point $x^{*}\in S_{x}$ such that $e^{(S_{n}f)(x^{*})}=\max \{ e^{(S_{n}f)(z)}: z\in S_{x}\}$ . Then we can construct a set $E_{n}(y)$ such that $x^{*}\in E_{n}(y)$ . In a similar manner, for each $x\in E_{n+m}(y)$ , define $S_{\sigma ^{n}x}$ by $S_{\sigma ^{n}x}:=\{x^{\prime }\in E_{n+m}(y): x^{\prime }_{i}=x_{i} , n+1\leq i\leq m+n\}$ and take a point $x^{**}\in S_{\sigma ^{n}x}$ such that $e^{(S_{m}f)(\sigma ^{n}x^{**})}=\max \{ e^{(S_{m}f)(z)}: z\in S_{\sigma ^{n}x}\}$ . Then we can construct a set $E_{m}(\sigma ^{n} y)$ such that $\sigma ^{n} x^{**}\in E_{m}(\sigma ^{n} y)$ . Hence, we obtain $g_{n+m}(y)\leq g_{n}(y)g_{m}(\sigma ^{n}y)$ . Next we show that ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ satisfies (D2). We modify slightly the arguments found in [14] (see also [12]) by taking account of the tempered variation of f, and we write a proof for completeness. Given $u\in B_{n}(Y)$ and $v \in B_{m}(Y)$ , let $x_{1}\ldots x_{n}\in B_{n}(X)$ such that $\pi (x_{1}\ldots x_{n})=u$ and let $z_{1}\ldots z_{m}\in B_{m}(X)$ such that $\pi (z_{1}\ldots z_{m})=v$ . Let p be a weak specification number of X. Then there exists $\tilde {w}\in B_{k}(X)$ , $0\leq k\leq p$ , such that $x_{1}\ldots x_{n}\tilde {w}z_{1}\ldots z_{m}\in B_{n+m+k}(X)$ . Hence, if $x\in [x_{1}\ldots x_{n} \tilde {w} z_{1}\ldots z_{m}]$ , by letting $\overline m=\min _{0\leq k\leq p}\{e^{(S_{k}f)(x)}:x\in X\}$ , where $e^{(S_{0}f)(x)}:=1$ for all $x\in X$ , we obtain

(34) $$ \begin{align} e^{(S_{n+k+m}f)(x)}\geq \overline {m} e^{(S_{n}f)(x)} e^{(S_{m}f)(\sigma^{n+k}x)}. \end{align} $$

For $n\in {\mathbb N}$ , let $M_{n}:=\sup \{e^{(S_{n}f)(x)}/e^{(S_{n}f)(x^{\prime })}: x_{i}=x^{\prime }_{i}, 1\leq i\leq n\}$ . Since X has the weak specification, Y also satisfies the weak specification property with specification number p. Define S by $S=\{w\in B_{k}(Y): 0\leq k\leq p, uwv\in B(Y)\}$ and let $y_{w}$ be a point from the cylinder set $[uwv]$ . Then

$$ \begin{align*} &\sum_{w\in S} \sum_{x\in E_{n+m+\vert w\vert}(y_{w})}e^{(S_{n+m+\vert w \vert}f)(x)}\geq \sum_{\substack{x\in [x_{1}\ldots x_{n}\tilde {w} z_{1}\ldots z_{m}],\\ \pi(x_{1}\ldots x_{n} \tilde {w} z_{1}\ldots z_{m})\in[uwv]}}\overline {m}e^{(S_{n}f)(x)} e^{(S_{m}f)(\sigma^{n+k}x)}\\ &\quad\geq \frac{\overline m}{M_{n}M_{m}}\bigg(\sum_{\pi(x_{1}\ldots x_{n})=u}\sup_{x\in [x_{1}\ldots x_{n}]}e^{(S_{n}f)(x)}\bigg) \bigg(\sum_{\pi(z_{1}\ldots z_{m})=v}\sup_{z\in [z_{1}\ldots z_{m}]}e^{(S_{m}f)(z)}\bigg) \\ &\quad\geq \frac{\overline m}{M_{n}M_{m}}\sup\{g_{n}(y): y\in[u]\}\sup\{g_{m}(y): y\in[v]\}. \end{align*} $$

Hence,

$$ \begin{align*} \sum_{w\in S}g_{n+m+\vert w\vert}(y_{w})\geq \frac{\overline {m}}{M_{n}M_{m}}\sup\{g_{n}(y): y\in[u]\}\sup\{g_{m}(y): y\in[v]\}. \end{align*} $$

Hence, there exits $\bar {w}\in S$ such that

$$ \begin{align*} g_{n+m+\vert \bar w\vert}(y_{\bar w})\geq \frac{\overline {m}}{M_{n}M_{m}\vert S\vert}\sup\{g_{n}(y): y\in[u]\}\sup\{g_{n}(y): y\in[v]\}. \end{align*} $$

If Y is a subshift on l symbols, then $\vert S\vert \leq l^{p}$ . Hence, ${\mathcal G}$ satisfies (D2) by setting $D_{n,m}=\overline {m}/(l^{p} M_{n}M_{m})$ . By the definition of ${\mathcal G}$ , clearly ${\mathcal G}$ has bounded variation.

Lemma 6.6. [28]

Let $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ be subshifts and $\pi :X\rightarrow Y$ be a one-block factor map. Given $f\in C(X)$ , the following statements are equivalent for $h\in C(Y)$ .

1. (i) $f-h\circ \pi $ is a compensation function for $\pi $ .

2. (ii) $\int P(\sigma _{X}, \pi , f-h\circ \pi )\,dm=0$ for each $m\in M(Y, \sigma _{Y})$ .

3. (iii) $m(\{y\in Y: P(\sigma _{X}, \pi , f-h\circ \pi )(y)=0\})=1$ for each $m\in M(Y, \sigma _{Y})$ .

Lemma 6.7. Let $(X, \sigma _{X})$ and $(Y, \sigma _{Y})$ be subshifts and $\pi :X\rightarrow Y$ be a one-block factor map. Given $f\in C(X)$ , the following statement for $h\in C(Y)$ is equivalent to the equivalent statements in Lemma 6.6:

$$ \begin{align*}\int P(\sigma_{X}, \pi, f) \,dm=\int h \,dm\ \mbox{for each}\ m\in M(Y, \sigma_{Y}).\end{align*} $$

Proof. Suppose that the equation in Lemma 6.7 holds for every $m \in M(Y,\sigma_Y )$ . Then (31) implies that $f-h\circ \pi $ is a compensation function for $\pi $ . Suppose that Lemma 6.6 (iii) holds. Then (33) implies that m-almost everywhere,

$$ \begin{align*} P(\sigma_{X}, \pi, f-h\circ\pi)(y)= \limsup_{n\rightarrow\infty} \frac{1}{n} \log \bigg(\frac{g_{n}(y)}{e^{(S_{n}h)(y)}}\bigg), \end{align*} $$

where $g_{n}(y)$ is defined as in (32). Since $\{\log g_{n}\}_{n=1}^{\infty }$ is subadditive, $\{\log (g_{n}/e^{S_{n}h})\}_{n=1}^{\infty }$ is subadditive. Applying the subadditive ergodic theorem, we obtain, for each $m\in M(Y, \sigma _{Y})$ ,

$$ \begin{align*} \begin{split} \int P(\sigma_{X}, \pi, f-h\circ \pi) \,dm &=\int \limsup_{n\rightarrow\infty} \frac{1}{n} \log \bigg (\frac{g_{n}}{e^{S_{n}h}} \bigg )\,dm \\&=\lim_{n\rightarrow\infty} \frac{1}{n} \int \log \bigg (\frac{g_{n}}{e^{S_{n}h}}\bigg )\,dm\\ &= \lim_{n\rightarrow\infty} \frac{1}{n} \int \log {g_{n}}\,dm -\int h \,dm=0. \end{split} \end{align*} $$

Hence, we obtain Lemma 6.7.

Lemma 6.8. Let $m\in M(Y, \sigma _{Y})$ . Then

$$ \begin{align*} P(\sigma_{X}, \pi, f-h\circ\pi)(y)= \lim_{n\rightarrow\infty} \frac{1}{n} \log \bigg (\frac{g_{n} (y)}{e^{(S_{n} h) (y)}} \bigg) \end{align*} $$

m-almost everywhere on Y.

Proof. The result follows by the subadditive ergodic theorem.

The main result of this section is the next theorem which relates the existence of a continuous compensation function for a factor map with the asymptotically additive property of the sequences ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ . Given $f\in C(X)$ , we continue to use $g_{n}$ as defined in equation (32).

Theorem 6.9. Let $(X,\sigma _{X})$ be an irreducible shift of finite type and $(Y,\sigma _{Y})$ be a subshift. Let $\pi : X\rightarrow Y$ be a one-block factor map and $f\in C(X)$ . Then the following statements are equivalent for $h\in C(Y)$ .

1. (i) $P(\sigma _{X}, \pi , f-h\circ \pi )(y)=0$ for every periodic point $y\in Y$ ; equivalently,

   $$ \begin{align*}\lim_{n\rightarrow\infty} \frac{1}{n} \log \bigg (\frac{g_{n} (y)}{e^{(S_{n} h) (y)}} \bigg)=0\end{align*} $$
   for every periodic point $y \in Y$ .

2. (ii) The function $f-h\circ \pi $ is a compensation function for $\pi $ .

3. (iii)

   $$ \begin{align*}\lim_{n\rightarrow\infty} \frac{1}{n} \log \bigg (\frac{g_{n} (y)}{e^{(S_{n} h) (y)}} \bigg)=0\end{align*} $$
   for every $y\in Y$ .

4. (iv) The sequence ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ is asymptotically additive on Y satisfying

   $$ \begin{align*}\lim_{n\rightarrow \infty} \frac{1}{n} \bigg\lVert \log \bigg (\frac{g_{n}}{e^{(S_{n} h)}}\bigg) \bigg\rVert_{\infty}=0.\end{align*} $$

5. (v) $\int P(\sigma _{X}, \pi , f)\,dm=\int h\,dm$ for all $m\in M(Y, \sigma _{Y})$ .

Remark 6.10.

1. (1) Theorem 6.9 (i) with $f=0$ is equivalent to the condition found by Shin [26, Theorem 3.5] for the existence of a saturated compensation function between two-sided irreducible shifts of finite type (see §7). Hence, by [26, Theorem 3.5], Theorem 6.9 (i), (ii) and (v) are equivalent when $f=0$ for a factor map between two-sided irreducible shifts of finite type. By the result of Cuneo [9, Theorem 2.1], if ${\mathcal G}$ is asymptotically additive then (v) holds for some $h\in C(Y)$ .

2. (2) See §7 for some examples and properties of h.

Proof. It is clear that (iv) implies (iii). By Lemma 6.6, (iii) implies (ii) and (ii) implies (i). Now we show that (i) implies (iv). Suppose that (i) holds. It is enough to show that Lemma 4.1 (ii) holds. Let X be an irreducible shift of finite type on a set S of finitely many symbols and k be a weak specification number of X. Let L be the cardinality of the set S. Let $y=(y_{1}, y_{2},\ldots , y_{n},\ldots )\in Y$ . For a fixed $n\geq 3$ , let $y_{1}=a$ , $y_{n}=b$ . Then $\pi ^{-1}(y_{1})=\{a_{1}, \ldots , a_{L_{1}}\}$ , where $a_{i}\in S$ for $1\leq i\leq L_{1}$ , for some $L_{1}\leq L$ , and $\pi ^{-1}(y_{n})=\{b_{1}, \ldots , b_{L_{2}}\}$ where $b_{j}\in S$ for $1\leq j\leq L_{2}$ , for some $L_{2}\leq L$ . Define $W_{ij}:=\{a_{i}x_{2}\ldots x_{n-1}b_{j}\in B_{n}(X): \pi (a_{i}x_{2}\ldots x_{n-1}b_{j})=y_{1}\ldots y_{n}\}$ . Let $E^{i,j}_{n}(y)$ be a set consisting of exactly one point from each cylinder set $[u]$ of length n of X, where $u\in W_{ij}$ . Define $C_{i,j}:=\sum _{x\in E^{i,j}_{n}(y)} e^{(S_{n}f)(x)}$ and $M_{n}:=\sup \{e^{(S_{n}f)(x)}/e^{(S_{n}f)(y)}: x_{i}=y_{i}, 1\leq i\leq n\}$ . If $W_{i,j}=\emptyset $ , then define $C_{i,j}:=0$ . Then

$$ \begin{align*} g_{n}(y)\geq \sum_{1\leq i\leq L_{1}, 1\leq j\leq L_{2}} C_{i,j}\geq \frac{1}{M_{n}}g_{n}(y), \end{align*} $$

where in the second equality we use the fact that, for any $E_{n}(y)$ ,

$$ \begin{align*} \frac{g_{n}(y)}{M_{n}} \leq \sum_{x\in E_{n}(y)} e^{(S_{n}f)(x)}. \end{align*} $$

Hence, there exist $i_{0}, j_{0}$ such that

(35) $$ \begin{align} C_{i_{0},j_{0}}\geq \frac{1}{L_{1}L_{2} M_{n}}g_{n}(y)\geq \frac{1}{L^{2} M_{n}}g_{n}(y). \end{align} $$

Note that $ (i_{0}, j_{0})$ depends on n. There exists an allowable word $w=w_{1}\ldots w_{q}$ of length q in X, $0\leq q \leq k$ , such that $b_{j_{0}}wa_{i_{0}}$ is an allowable word of X. Take an allowable word $a_{i_{0}}x_{2}\ldots x_{n-1}b_{j_{0}}\in W_{i_{0}, j_{0}}$ . Since X is an irreducible shift of finite type, we obtain a periodic point $\tilde {x}:=(a_{i_{0}},x_{2},\ldots , x_{n-1},b_{j_{0}}, w_{1},\ldots , w_{q})^{\infty }\in X$ . Let $\pi (w_{i})=d_{i}$ for each $i=1, \ldots , q$ . Let $y^{*}:=\pi (\tilde {x})$ . Then $y^{*}=(y_{1},\ldots , y_{n},d_{1},\ldots d_{q})^{\infty }$ is a periodic point of $\sigma _{Y}$ .

For a fixed $n\geq 3$ , define $P_{0}:=E^{i_{0},j_{0}}_{n}(y)$ . Define $P_{1}$ by

$$ \begin{align*} P_{1}=\{z=(z_{i})_{i=1}^{\infty}\in X: z_{1}\ldots z_{n} \in W_{i_{0}, j_{0}}, z_{n+1}\ldots z_{n+q}=w, \sigma^{n+q} z=z \}. \end{align*} $$

Observe that if $z\in P_{1}$ , then $\pi (z)=y^{*}$ and $P_{1}$ is a set consisting of exactly one point from each cylinder $[u]$ of length $(n+q)$ of X such that $\pi (u)= y_{1}\ldots y_{n}d_{1}\ldots d_{q}$ satisfying $u_{1} \ldots u_{n} \in W_{i_{0}, j_{0}}$ and $u_{n+1}\ldots u_{n+q}=w$ . Then

$$ \begin{align*} g_{n+q} (y^{*})=\sup_{E_{n+q}(y^{*})}\bigg\{\sum_{x\in E_{n+q}(y^{*})}e^{(S_{n+q}f)(x)}\bigg\} \geq \sum_{x\in P_{1}}e^{(S_{n+q}f)(x)} \geq \frac{e^{m}}{M_{n}}\bigg( \sum_{x\in P_{0}}e^{(S_{n}f)(x)}\bigg), \end{align*} $$

where $m:= \min _{0\leq i\leq k}\{e^{(S_{i}f)(x)}: x\in X\}$ , $(S_{0}f)(x):=1$ for every $x\in X$ . Next define $P_{2}$ by

$$ \begin{align*} \begin{split} P_{2}&=\{z=(z_{i})_{i=1}^{\infty}\in X: \ \text{for each } j=0,1, z_{j(n+q)+1}\ldots z_{n(j+1)+jq} \in W_{i_{0}, j_{0}},\\ &\ \quad z_{(j+1)n+jq+1}\ldots z_{(j+1)(n+q)}=w, \sigma^{2(n+q)} z=z \}. \end{split} \end{align*} $$

Observe that if $z\in P_{2}$ , then $\pi (z)=y^{*}$ and $P_{2}$ is a set consisting of one point from each cylinder $[u]$ of length $(2n+2q)$ of X such that $\pi (u)= y_{1}\ldots y_{n}d_{1}\ldots d_{q}y_{1}\ldots y_{n}d_{1}\ldots d_{q}$ satisfying $u_{1}\ldots u_{n}, u_{n+q+1}\ldots u_{2n+q}\in W_{i_{0}, j_{0}}$ and $u_{n+1}\ldots u_{n+q}=u_{2n+q+1}\ldots u_{2n+2q}=w$ . Hence,

$$ \begin{align*} g_{2(n+q)} (y^{*})&=\sup_{E_{2(n+q)}(y^{*})}\bigg\{\sum_{x\in E_{2(n+q)}(y^{*})}e^{(S_{2(n+q)}f)(x)}\bigg\}\\ &\geq \sum_{x\in P_{2}}e^{(S_{2(n+q)}f)(x)} \geq \frac{e^{2m}}{M^{2}_{n}}\bigg( \sum_{x\in P_{0}}e^{(S_{n}f)(x)}\bigg)^{2}. \end{align*} $$

Applying (35), we obtain

$$ \begin{align*} g_{2(n+q)} (y^{*})\geq \frac{e^{2m}}{M^{2}_{n}}\bigg(\sum_{x\in P_{0}}e^{(S_{n}f)(x)}\bigg)^{2} \geq \frac{e^{2m}g^{2}_{n}(y^{*})}{L^{4}M^{4}_{n}}. \end{align*} $$

Similarly, for $j\geq 3$ , define the set $P_{j}$ of periodic points by

$$ \begin{align*} \begin{split} P_{j}&=\{z=(z_{i})_{i=1}^{\infty}\in X: \quad\text{for each } 0\leq l \leq j-1, z_{l(n+q)+1}\ldots z_{(l+1)n+lq}\in W_{i_{0}, j_{0}}, \\ &\qquad z_{(l+1)n+lq+1}\ldots z_{(l+1)(n+q)}=w,\sigma^{j(n+q)} z=z \}. \end{split} \end{align*} $$

If $z\in P_{j}$ , then $\pi (z)=y^{*}$ and $P_{j}$ is a set consisting of one point from each cylinder $[u]$ of length $j(n+q)$ such that $\pi (u)= (y_{1}\ldots y_{n}d_{1}\ldots d_{q})^{j}$ satisfying $u_{l(n+q)+1}=a_{i_{0}}$ , $u_{(l+1)n+lq}=b_{j_{0}}$ and $u_{(l+1)n+lq+1}\ldots u_{(l+1)(n+q)}=w$ for each $0\leq l\leq j-1$ . Then we obtain

$$ \begin{align*} g_{j(n+q)}(y^{*})&=\sup_{E_{j(n+q)}(y^{*})}\bigg\{\sum_{x\in E_{j(n+q)}(y^{*})}e^{(S_{j(n+q)}f)(x)}\bigg\}\\ &\geq \sum_{x\in P_{j}}e^{(S_{j(n+q)}f)(x)} \geq \frac{e^{jm}}{M^{j}_{n}} \bigg( \sum_{x\in P_{0}}e^{(S_{n}f)(x)}\bigg)^{j}. \end{align*} $$

Applying (35), we obtain

$$ \begin{align*} g_{j(n+q)}(y^{*})\geq \frac{e^{jm}}{M^{j}_{n}} \bigg(\sum_{x\in P_{0}}e^{(S_{n}f)(x)}\bigg)^{j} \geq \bigg(\frac{e^{m}}{L^{2}{M^{2}_{n}}}\bigg)^{j}g^{j}_{n}(y^{*}). \end{align*} $$

Since the function $g_{n}$ is locally constant, for $n\geq 3$ ,

$$ \begin{align*} g_{j(n+q)}(y^{*})\geq \bigg(\frac{e^{m}}{L^{2}{M^{2}_{n}}}\bigg)^{j}\sup\{g_{n}(z): z\in [y_{1}\ldots y_{n}]\}^{j} \end{align*} $$

for every $j\in {\mathbb N}$ . Hence, condition (ii) in Lemma 4.1 holds. Applying Lemma 4.1, we obtain (iv). Finally, (iv) ${\implies}$ (v) is immediate and (v) implies (ii) by Lemma 6.7.

Recall that if $f\in C(X)$ is in the Bowen class, then ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ satisfies (C1) and (C2) and ${\mathcal G}$ has the unique Gibbs equilibrium state.

Corollary 6.11. Under the assumptions of Theorem 6.9, assume also that $f\in C(X)$ is a function in the Bowen class and let m be the unique Gibbs equilibrium state for ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ . Suppose that one of the equivalent statements in Theorem 6.9 holds. Then:

1. (i) m is an invariant weak Gibbs measure for h;

2. (ii) equation (28) holds by replacing $f_{n}$ by $g_{n}$ ;

3. (iii) if m is a Gibbs measure for a continuous function, then ${\mathcal G}$ is almost additive.

Hence, if there is no sequence $\{C_{n, m}\}_{n,m\in {\mathbb N}}$ satisfying (28) by replacing $f_{n}$ by $g_{n}$ , then there does not exist a continuous function h on Y such that

$$ \begin{align*}\int P(\sigma_{X}, \pi, f) \,d\kern-1pt\mu=\int h \,d\kern-1pt\mu\end{align*} $$

for every $\mu \in M(Y, \sigma _{Y}).$

Proof. Since $\lim _{n\rightarrow \infty }(1/n)\vert \vert \log (g_{n}/e^{S_{n}h})\vert \vert _{\infty }=0$ , applying the first part of the proof of Lemma 5.5, m is an invariant weak Gibbs measure for h. To show the second statement, we use similar arguments to the proof of Theorem 5.6. To show the third statement, we apply the proof of Theorem 5.6. The last statement is obvious by Theorem 6.9.

Remark 6.12. Applying Theorem 4.3, we can study Theorem 6.9 under a more general setting. Let $(X,\sigma _{X}), (Y,\sigma _{Y})$ be subshifts and $\pi : X\rightarrow Y$ be a one-block factor map. Given a function $f\in C(X)$ , suppose that ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ satisfies Theorem 4.4 (ii). Then Theorem 6.9 holds. It would be interesting to study the conditions on factor maps $\pi $ satisfying Theorem 4.4 (ii) for ${\mathcal G}$ .

7 Applications

In this section, we give some examples and applications. Applying the results from the previous sections, we study the existence of a saturated compensation function for a factor map between subshifts and factors of weak Gibbs measures for continuous functions.

7.1 Existence of continuous saturated compensation functions

Let $(X,\sigma _{X})$ be an irreducible shift of finite type, Y be a subshift and $\pi : X\rightarrow Y$ be a one-block factor map. For $n\in {\mathbb N}$ , let $\phi _{n}$ be the continuous function on Y obtained by setting $f=0$ in equation (32). Set $\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }$ .

For a factor map $\pi $ between subshifts, there always exists a Borel measurable saturated compensation function $-P(\sigma _{X}, \pi , 0)\circ \pi $ given by a superadditive sequence $-\Phi \circ \pi $ ; however, a continuous saturated compensation function does not always exist. Shin [26] considered a one-block factor map $\pi : X\rightarrow Y$ between two-sided irreducible shifts of finite type and gave an equivalent condition for the existence of a saturated compensation function (see [26, Theorem 3.5] for details). Note that the condition is equivalent to Theorem 6.9 (i) with $f=0$ .

Here we characterize the existence of a saturated compensation function in terms of the type of the sequence $\Phi $ by applying Theorem 6.9.

Corollary 7.1. Let $(X,\sigma _{X})$ be an irreducible shift of finite type, Y be a subshift and $\pi : X\rightarrow Y$ be a one-block factor map. Then $-h\circ \pi , h\in C(Y)$ is a saturated compensation function if and only if one of the equivalent statements in Theorem 6.9 holds with $f=0$ . In particular, a saturated compensation function exists if and only if $\Phi $ is asymptotically additive on Y. If $-h\circ \pi $ is a compensation function, then h has the unique equilibrium state and it is a weak Gibbs measure for h. If there does not exist $\{C_{n,m}\}_{(n,m)\in {\mathbb N} \times {\mathbb N}}$ satisfying equation (28) for $\Phi $ , then there exists no continuous saturated compensation function for $\pi $ .

Proof. The result follows by setting $f=0$ in Theorem 6.9 and Corollary 6.11.

Example 7.2. (A sequence satisfying (C1) and (C2) which is not asymptotically additive [26])

Shin [26, Example 3.1] gave an example of a factor map $\pi : X\rightarrow Y$ between two-sided irreducible shifts of finite type $X,Y$ without a saturated compensation function. We note that the same results hold for one-sided subshifts. The sequence $\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }$ is a subadditive sequence satisfying (C1) and (C2) with bounded variation and there exists a unique Gibbs equilibrium state $\nu $ for $\Phi $ . Since there is no saturated compensation function, there does not exist a continuous function $h\in C(Y)$ such that

$$ \begin{align*}\lim_{n\rightarrow\infty}\frac{1}{n}\int\log \phi_{n}\,dm=\int h\,dm\end{align*} $$

for every $m\in M(Y, \sigma _{Y})$ . Hence, $\Phi $ is not an asymptotically additive sequence and there does not exist a continuous function on Y for which $\nu $ is an invariant weak Gibbs measure (see Theorem 7.8). Alternatively, a simple calculation shows that for any $x\in [12^{m}1]$ where $m\geq 3$ is odd,

$$ \begin{align*}\frac{\phi_{2+m}(x)}{\phi_{2}(x)\phi_{m}(\sigma^{2}x)}=\frac{\vert \pi^{-1}[12^{m}1]\vert} {\vert\pi^{-1}[12]\vert \vert \pi^{-1}[2^{m-1}1]\vert}=\frac{1}{2^{({m-1})/{2}}+2}\end{align*} $$

(see [26]). Hence, for any sequence $\{C_{n,m}\}_{(n,m)\in {\mathbb N} \times {\mathbb N}}$ satisfying equation (28) for $\Phi $ , we obtain that $C_{2, m}\geq {2^{({m-1})/{2}}+2}$ . By Corollary 7.1, there does not exist a continuous saturated compensation function.

Remark 7.3.

1. (1) Pfister and Sullivan [20] studied a class of continuous functions satisfying bounded total oscillations on two-sided subshifts and showed that if a continuous function f belongs to the class under a certain condition then an equilibrium state for f is a weak Gibbs measure for some continuous function. Shin [24, Proposition 3.5] gave an example of a saturated compensation function $G\circ \pi $ for a factor map $\pi : X\rightarrow Y$ between two-sided irreducible shifts of finite type $X,Y$ where $-G$ does not have bounded total oscillations. Let $(X^{+}, \sigma ^{+}_{X})$ and $(Y^{+}, \sigma ^{+}_{Y})$ be the corresponding one-sided shifts of finite type and consider the factor map $\pi ^{+}: X^{+}\rightarrow Y^{+}$ . Then the corresponding saturated compensation function $G^{+}\circ \pi $ for $\pi ^{+}, G^{+}\in C(Y^{+}),$ is obtained. Applying Theorem 6.9 and Corollary 6.11, $-G^{+}$ has a unique equilibrium state and it is a weak Gibbs measure for $-G^{+}$ .

2. (2) See §2 in [3] for examples of measures which are not weak Gibbs studied in quantum physics.

Example 7.4. (A sequence satisfying (C1) and (C2) which is also asymptotically additive)

In [30], saturated compensation functions were studied to find the Hausdorff dimensions of some compact invariant sets of expanding maps of the torus. In [30, Example 5.1], given a factor map $\pi $ between topologically mixing shifts of finite type X and Y, a saturated compensation function $G\circ \pi $ , $G\in C(Y)$ , was found and $-G$ has a unique equilibrium state $\nu $ which is not Gibbs. Applying Theorem 6.9 and Corollary 6.11, $\nu $ is an invariant weak Gibbs measure for $-G$ .

Remark 7.5.

1. (1) In [12, 31], the ergodic measures of full Hausdorff dimension for some compact invariant sets of certain expanding maps of the torus were identified with equilibrium states for sequences of continuous functions. If a saturated compensation function exists, then they are the equilibrium states of a constant multiple of a saturated compensation [30].

2. (2) In Example 7.4 [30, Example 5.1], X and Y are one-sided shifts of finite type. Considering the corresponding two-sided shifts of finite type $\hat X, \hat Y$ and the factor map $\hat \pi $ between them, a saturated compensation function $\hat G\circ \pi $ for $\hat \pi $ , $\hat G\in C(\hat Y)$ , is obtained in the same manner as G is obtained. The function $-\hat G$ on $\hat Y$ does not have bounded total oscillations (see Remark 7.3(1)).

7.2 Factors of invariant weak Gibbs measures

Factors of invariant Gibbs measures for continuous functions and related topics have been widely studied (see, for example, [7, 8, 12, 16, 21–23, 27, 31–33]). For a survey of the study of factors of Gibbs measures, see the paper by Boyle and Petersen [4]. In this section, more generally, we study the properties of factors of invariant weak Gibbs measures. Given a one-block factor map $\pi : X\rightarrow Y$ , and $f\in C(X)$ , define $g_{n}$ for each $n\in {\mathbb N}$ as in (32) and ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty } $ on Y.

Theorem 7.6. Let $(X,\sigma _{X})$ be an irreducible shift of finite type, Y be a subshift and $\pi : X\rightarrow Y$ be a one-block factor map. Suppose there exists $\mu $ such that $\mu $ is an invariant weak Gibbs measure for $f\in C(X)$ . Then $\pi \mu $ is an invariant weak Gibbs measure for ${\mathcal G}=\{\log g_{n}\}_{n=1}^{\infty }$ on Y. There exists $h\in C(Y)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log g_{n} \,dm =\int h \,dm$ for all $m\in M(Y,\sigma _{Y})$ if and only if one of the equivalent statements in Theorem 6.9(i)–(iv) holds. Moreover, such a function h exists if and only if the invariant measure $\pi \mu $ is a weak Gibbs measure for a continuous function on Y.

Remark 7.7.

1. (1) If f is in the Bowen class, then there is a unique Gibbs equilibrium state for ${\mathcal G}$ and Corollary 6.11 also applies.

2. (2) If there exists $\mu $ such that $\mu $ is an invariant weak Gibbs measure for $f\in C(X)$ , then $\pi \mu $ is an equilibrium state for ${\mathcal G}$ .

Proof. To prove the first statement, we apply similar arguments to the proof of [32, Theorem 3.7] and we outline the proof. Suppose that $f\in C(X)$ has an invariant weak Gibbs measure $\mu $ . Then there exists $C_{n}>0$ such that

$$ \begin{align*} \frac{1}{C_{n}} \leq \frac{\mu [x_{1}\ldots x_{n}]}{e^{-nP(f)+(S_{n}f)(x)}} \leq {C_{n}} \end{align*} $$

for each $x\in [x_{1}\ldots x_{n}]$ , where $\lim _{n\rightarrow \infty }(1/n)\log C_{n}=0$ . Since f has tempered variation, if we let

$$ \begin{align*}M_{n}=\sup\bigg\{\frac{e^{(S_{n}f)(x)}}{e^{(S_{n}f)(y)}}:x,y \in X, x_{i}=y_{i} \text{ for } 1 \leq i \leq n\bigg\},\end{align*} $$

then $\lim _{n\rightarrow \infty }(1/n)\log M_{n}=0$ . Using the definition of the topological pressure and after some calculations, we obtain that $P(f)=P({\mathcal G})$ . Since

$$ \begin{align*}\pi\mu[y_{1}\ldots y_{n}]=\sum_{\substack{x_{1}\ldots x_{n}\in B_{n}(X)\\ \pi(x_{1}\ldots x_{n})=y_{1}\ldots y_{n}}}\mu[x_{1}\ldots x_{n}],\end{align*} $$

using similar arguments to the proof of [32, Theorem 3.7], we obtain

$$ \begin{align*} \frac{1}{C_{n}M_{n}} \leq \frac{\pi\mu [y_{1}\ldots y_{n}]}{e^{-nP({\mathcal G})}g_{n}(y)} \leq {C_{n}M_{n}}. \end{align*} $$

Hence, $\pi \mu $ is an invariant weak Gibbs measure for ${\mathcal G}$ . The second statement holds by Theorem 6.9. Now we show the last statement. Suppose such h exists. Modifying slightly the proof of Corollary 6.11 (i), taking into account the fact that $\pi \mu $ is a weak Gibbs measure for ${\mathcal G}$ , we obtain that $\pi \mu $ is a weak Gibbs measure for h. To see the reverse implication, suppose $\pi \mu $ is weak Gibbs for some $\tilde {h}$ . Then there exists $A_{n}>0$ such that

(36) $$ \begin{align} \frac{1}{A_{n}} \leq \frac{\pi\mu [y_{1}\ldots y_{n}]}{e^{-nP(\tilde h)+(S_{n}\tilde h)(y)}} \leq {A_{n}} \end{align} $$

for each $y\in [y_{1}\ldots y_{n}]$ , where $\lim _{n\rightarrow \infty }(1/n)\log A_{n}=0$ . If we let $K_{n}=C_{n}M_{n}$ , then the similar arguments to the latter part of the proof of Lemma 5.5 show that

$$ \begin{align*}\frac{1}{K_{n}A_{n}}\leq \frac{g_{n}(y)}{e^{(S_{n}(\tilde{h}-P(\tilde{h})+P({\mathcal G})))(y)}}\leq K_{n}A_{n}\end{align*} $$

for each $y\in [y_{1}\ldots y_{n}]$ . Hence, ${\mathcal G}$ is asymptotically additive. Set $h=\tilde h- P(\tilde {h}) +P({\mathcal G}).$

The proof of Theorem 7.6 gives us the following result.

Theorem 7.8. Under the assumptions of Theorem 6.9, suppose there exists $\mu $ such that $\mu $ is an invariant weak Gibbs measure for $f\in C(X)$ . Then there exists $h\in C(Y)$ satisfying the equivalent statements in Theorem 6.9 if and only if there exists a continuous function on Y for which $\pi \mu $ is an invariant weak Gibbs measure on Y.

Corollary 7.9. Under the assumptions of Theorem 7.6, if there is no sequence $\{C_{n,m}\}_{n, m\in {\mathbb N}}$ satisfying equation (28) by replacing $f_{n}$ by $g_{n}$ , then there does not exist a continuous function h on Y such that $\lim _{n\rightarrow \infty } (1/{n})\int \log g_{n} \,dm=\int h \,dm$ for every $m\in M(Y, \sigma _{Y})$ . Hence, there exists no continuous function on Y for which $\pi \mu $ is an invariant weak Gibbs measure on Y.

Proof. Suppose there exists $h\in C(Y)$ such that $\lim _{n\rightarrow \infty } (1/{n})\int \log g_{n} \,dm=\int h \,dm$ for every $m\in M(Y, \sigma _{Y})$ . By Theorem 7.6, ${\mathcal G}$ is asymptotically additive and $\pi \mu $ is an invariant weak Gibbs measure for h. Hence, there exists $A_{n}>0$ such that (36) holds for h for each $y\in [y_{1}\ldots y_{n}]$ , where $\lim _{n\rightarrow \infty }(1/n)\log A_{n}=0$ . Let $K_{n}$ be defined as in the proof of Theorem 7.6. Using $P(h)=P({\mathcal G})$ and additivity of $\{S_{n} h\}_{n=1}^{\infty }$ , we obtain

$$ \begin{align*}\frac{1}{K_{n+m}A_{n+m}K_{n}A_{n}K_{m}A_{m}}\leq \frac{g_{n+m}(y)}{g_{n}(y) g_{m}(\sigma^{n}_{Y} y)}\leq K_{n+m}A_{n+m}K_{n}A_{n}K_{m}A_{m}.\end{align*} $$

Define $C_{n,m}:= K_{n+m}A_{n+m}K_{n}A_{n}K_{m}A_{m}$ for each $n,m \in {\mathbb N}$ . Then $\lim _{n\rightarrow \infty }(1/n)\log C_{n,m}=\lim _{m\rightarrow \infty }(1/m)\log C_{n,m}=0$ . Hence, the result follows from Theorem 7.6.