Phase transitions for non-singular Bernoulli actions

Abstract

Inspired by the phase transition results for non-singular Gaussian actions introduced in [AIM19], we prove several phase transition results for non-singular Bernoulli actions. For generalized Bernoulli actions arising from groups acting on trees, we are able to give a very precise description of their ergodic-theoretical properties in terms of the Poincaré exponent of the group.

1 Introduction

When G is a countable infinite group and $(X_0,\mu _0)$ is a non-trivial standard probability space, the probability measure-preserving (pmp) action

$$ \begin{align*} G\curvearrowright (X_0,\mu_0)^{G}:\quad (g\cdot x)_h=x_{g^{-1}h} \end{align*} $$

is called a Bernoulli action. Probability measure-preserving Bernoulli actions are among the best-studied objects in ergodic theory and they play an important role in operator algebras [Ioa10, Pop03, Pop06]. When we consider a family of probability measures $(\mu _g)_{g\in G}$ on the base space $X_0$ that need not all be equal, the Bernoulli action

(1.1) $$ \begin{align} G\curvearrowright (X,\mu)=\prod_{g\in G}(X_0,\mu_g) \end{align} $$

is in general no longer measure-preserving. Instead, we are interested in the case where $G\curvearrowright (X,\mu )$ is non-singular, that is, the group G preserves the measure class of $\mu $ . By Kakutani’s criterion for equivalence of infinite product measures the Bernoulli action (1.1) is non-singular if and only if $\mu _h\sim \mu _g$ for every $h,g\in G$ and

(1.2) $$ \begin{align} \sum_{h\in G}H^2(\mu_h,\mu_{gh})<+\infty\quad\text{for every } g\in G. \end{align} $$

Here $H^2(\mu _h,\mu _{gh})$ denotes the Hellinger distance between $\mu _h$ and $\mu _{gh}$ (see (2.2)).

It is well known that a pmp Bernoulli action $G\curvearrowright (X_0,\mu _0)^{G}$ is mixing. In particular, it is ergodic and conservative. However, for non-singular Bernoulli actions, determining conservativeness and ergodicity is much more difficult (see, for instance, [BKV19, Dan18, Kos18, VW17]).

Besides non-singular Bernoulli actions, another interesting class of non-singular group actions comes from the Gaussian construction, as introduced in [AIM19]. If ${\pi \colon G\rightarrow \mathcal {O}(\mathcal {H})}$ is an orthogonal representation of a locally compact second countable (lcsc) group on a real Hilbert space $\mathcal {H}$ , and if $c\colon G\rightarrow \mathcal {H}$ is a 1-cocycle for the representation $\pi $ , then the assignment

(1.3) $$ \begin{align} \alpha_g(\xi)=\pi_g(\xi)+c(g) \end{align} $$

defines an affine isometric action $\alpha \colon G\curvearrowright \mathcal {H}$ . To any affine isometric action $\alpha \colon G\curvearrowright \mathcal {H}$ Arano, Isono and Marrakchi associated a non-singular group action $\widehat {\alpha }\colon G\curvearrowright \widehat {\mathcal {H}}$ , where $\widehat {\mathcal {H}}$ is the Gaussian probability space associated to $\mathcal {H}$ . When $\alpha \colon G\curvearrowright \mathcal {H}$ is actually an orthogonal representation, this construction is well established and the resulting Gaussian action is pmp. As explained below [BV20, Theorem D], if G is a countable infinite group and $\pi \colon G\rightarrow \ell ^2(G)$ is the left regular representation, the affine isometric representation (1.3) gives rise to a non-singular action that is conjugate with the Bernoulli action $G\curvearrowright \prod _{g\in G}(\mathbb {R},\nu _{F(g)})$ , where $F\colon G\rightarrow \mathbb {R}$ is such that $c_g(h)=F(g^{-1}h)-F(h)$ , and $\nu _{F(g)}$ denotes the Gaussian probability measure with mean $F(g)$ and variance $1$ .

By scaling the 1-cocycle $c\colon G\rightarrow \mathcal {H}$ with a parameter $t\in [0,+\infty )$ we get a one-parameter family of non-singular actions $\widehat {\alpha }^{t}\colon G\curvearrowright \widehat {\mathcal {H}}^{t}$ associated to the affine isometric actions $\alpha ^{t}\colon G\curvearrowright \mathcal {H}$ , given by $\alpha ^t_g(\xi )=\pi _g(\xi )+tc(g)$ . Arano, Isono and Marrakchi showed that there exists a $t_{\mathrm {diss}}\in [0,+\infty )$ such that $\widehat {\alpha }^t$ is dissipative up to compact stabilizers for every $t>t_{\mathrm {diss}}$ and infinitely recurrent for every $t<t_{\mathrm {diss}}$ (see §2 for terminology).

Inspired by the results obtained in [AIM19], we study a similar phase transition framework, but in the setting of non-singular Bernoulli actions. Such a phase transition framework for non-singular Bernoulli actions was already considered by Kosloff and Soo in [KS20]. They showed the following phase transition result for the family of non-singular Bernoulli actions of $G=\mathbb {Z}$ with base space $X_0=\{0,1\}$ that was introduced in [VW17, Corollary 6.3]. For every $t\in [0,+\infty )$ consider the family of measures $(\mu _n^t)_{n\in \mathbb {Z}}$ given by

$$ \begin{align*} \mu_n^t(0)=\begin{cases}1/2 &\text{ if } n\leq 4t^2,\\ 1/2+t/\sqrt{n}&\text{ if }n>4t^2.\end{cases} \end{align*} $$

Then $\mathbb {Z}\curvearrowright (X,\mu _t)=\prod _{n\in \mathbb {Z}}(\{0,1\},\mu _n^t)$ is non-singular for every $t\in [0,+\infty )$ . Kosloff and Soo showed that there exists a $t_1\in (1/6,+\infty )$ such that $ \mathbb {Z}\curvearrowright (X,\mu _t)$ is conservative for every $t<t_1$ and dissipative for every $t>t_1$ [KS20, Theorem 3]. In [DKR20, Example D] the authors describe a family of non-singular Poisson suspensions for which a similar phase transition occurs. These examples arise from dissipative essentially free actions of  $\mathbb {Z}$ , and thus they are non-singular Bernoulli actions. We generalize the phase transition result from [KS20] to arbitrary non-singular Bernoulli actions as follows.

Suppose that G is a countable infinite group and let $(\mu _g)_{g\in G}$ be a family of equivalent probability measure on a standard Borel space $X_0$ . Let $\nu $ also be a probability measure on $X_0$ . For every $t\in [0,1]$ we consider the family of equivalent probability measures $(\mu _g^t)_{g\in G}$ that are defined by

(1.4) $$ \begin{align} \mu_g^t=(1-t)\nu+t\mu_g. \end{align} $$

Our first main result is that in this setting there is a phase transition phenomenon.

Theorem A. Let G be a countable infinite group and assume that the Bernoulli action $G\curvearrowright (X,\mu _1)=\prod _{g\in G}(X_0,\mu _g)$ is non-singular. Let $\nu \sim \mu _e$ be a probability measure on $X_0$ and for every $t\in [0,1]$ consider the family $(\mu _g^t)_{g\in G}$ of equivalent probability measures given by (1.4). Then the Bernoulli action

$$ \begin{align*} G\curvearrowright (X,\mu_t)=\prod_{g\in G}(X_0,\mu_g^t) \end{align*} $$

is non-singular for every $t\in [0,1]$ and there exists a $t_1\in [0,1]$ such that $G\curvearrowright (X,\mu _t)$ is weakly mixing for every $t<t_1$ and dissipative for every $t>t_1$ .

Suppose that G is a non-amenable countable infinite group. Recall that for any standard probability space $(X_0,\mu _0)$ , the pmp Bernoulli action $G\curvearrowright (X_0,\mu _0)^{G}$ is strongly ergodic. Consider again the family of probability measures $(\mu _g^t)_{g\in G}$ given by (1.4). In Theorem B below we prove that for t close enough to $0$ , the resulting non-singular Bernoulli action is strongly ergodic. This is inspired by [AIM19, Theorem 7.20] and [MV20, Theorem 5.1], which state similar results for non-singular Gaussian actions.

Theorem B. Let G be a countable infinite non-amenable group and suppose that the Bernoulli action $G\curvearrowright (X,\mu _1)=\prod _{g\in G}(X_0,\mu _g)$ is non-singular. Let $\nu \sim \mu _e$ be a probability measure on $X_0$ and for every $t\in [0,1]$ consider the family $(\mu _g^t)_{g\in G}$ of equivalent probability measures given by (1.4). Then there exists a $t_0\in (0,1]$ such that $G\curvearrowright (X,\mu _t)=\prod _{g\in G}(X_0,\mu _g^t)$ is strongly ergodic for every $t<t_0$ .

Although we can prove a phase transition result in large generality, it remains very challenging to compute the critical value $t_1$ . However, when $G\subset \operatorname {Aut}(T)$ , for some locally finite tree T, following [AIM19, §10], we can construct generalized Bernoulli actions of which we can determine the conservativeness behaviour very precisely. To put this result into perspective, let us first explain briefly the construction from [AIM19, §10].

For a locally finite tree T, let $\Omega (T)$ denote the set of orientations on T. Let $p\in (0,1)$ and fix a root $\rho \in T$ . Define a probability measure $\mu _p$ on $\Omega (T)$ by orienting an edge towards $\rho $ with probability p and away from $\rho $ with probability $1-p$ . If $G\subset \operatorname {Aut}(T)$ is a subgroup, then we naturally obtain a non-singular action $G\curvearrowright (\Omega (T),\mu _p)$ . Up to equivalence of measures, the measure $\mu _p$ does not depend on the choice of root $\rho \in T$ . The Poincaré exponent of $G\subset \operatorname {Aut}(T)$ is defined as

(1.5) $$ \begin{align} \delta(G\curvearrowright T)=\inf\left\{s>0 \text{ for which }\sum_{w\in G\cdot v}\exp(-s d(v,w))<+\infty\right\}, \end{align} $$

where $v\in V(T)$ is any vertex of T. In [AIM19, Theorem 10.4] Arano, Isono and Marrakchi showed that if $G\subset \operatorname {Aut}(T)$ is a closed non-elementary subgroup, the action $G\curvearrowright (\Omega (T),\mu _p)$ is dissipative up to compact stabilizers if $2\sqrt {p(1-p)}<\exp (-\delta )$ and weakly mixing if $2\sqrt {p(1-p)}>\exp (-\delta )$ . This motivates the following similar construction.

Let $E(T)\subset V(T)\times V(T)$ denote the set of oriented edges, so that vertices v and w are adjacent if and only if $(v,w),(w,v)\in E(T)$ . Suppose that $X_0$ is a standard Borel space and that $\mu _0,\mu _1$ are equivalent probability measures on $X_0$ . Fix a root $\rho \in T$ and define a family of probability measures $(\mu _e)_{e\in E(T)}$ by

(1.6) $$ \begin{align} \mu_e=\begin{cases}\mu_0 &\text{if } e \text{ is oriented towards } \rho,\\ \mu_1 &\text{if } e \text{ is oriented away from } \rho.\end{cases} \end{align} $$

Suppose that $G\subset \operatorname {Aut}(T)$ is a subgroup. Then the generalized Bernoulli action

(1.7) $$ \begin{align} G\curvearrowright \prod_{e\in E(T)}(X_0,\mu_e):\quad (g\cdot x)_e=x_{g^{-1}\cdot e} \end{align} $$

is non-singular and up to conjugacy it does not depend on the choice of root $\rho \in T$ . In our next main result we generalize [AIM19, Theorem 10.4] to non-singular actions of the form (1.7).

Theorem C. Let T be a locally finite tree with root $\rho \in T$ and let $G\subset \operatorname {Aut}(T)$ be a non-elementary closed subgroup with Poincaré exponent $\delta =\delta (G\curvearrowright T)$ . Let $\mu _0$ and $\mu _1$ be equivalent probability measures on a standard Borel space $X_0$ and define a family of equivalent probability measures $(\mu _e)_{e\in E(T)}$ by (1.6). Then the generalized Bernoulli action (1.7) is dissipative up to compact stabilizers if $1-H^2(\mu _0,\mu _1)<\exp (-\delta /2)$ and weakly mixing if $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ .

2 Preliminaries

2.1 Non-singular group actions

Let $(X,\mu ), (Y,\nu )$ be standard measure spaces. A Borel map $\varphi \colon X\rightarrow Y$ is called non-singular if the pushforward measure $\varphi _*\mu $ is equivalent to $\nu $ . If in addition there exist conull Borel sets $X_0\subset X$ and $Y_0\subset Y$ such that $\varphi \colon X_0\rightarrow Y_0$ is a bijection we say that $\varphi $ is a non-singular isomorphism. We write $\operatorname {Aut}(X,\mu )$ for the group of all non-singular automorphisms $\varphi \colon X\rightarrow X$ , where we identify two elements if they agree almost everywhere. The group $\operatorname {Aut}(X,\mu )$ carries a canonical Polish topology.

A non-singular group action $G\curvearrowright (X,\mu )$ of an lcsc group G on a standard measure space $(X,\mu )$ is a continuous group homomorphism $G\rightarrow \operatorname {Aut}(X,\mu )$ . A non-singular group action $G\curvearrowright (X,\mu )$ is called essentially free if the stabilizer subgroup $G_x=\{g\in G:g\cdot x=x\}$ is trivial for almost every (a.e.) $x\in X$ . When G is countable this is the same as the condition that $\mu (\{x\in X:g\cdot x=x\})=0$ for every $g\in G\setminus \{e\}$ . We say that $G\curvearrowright (X,\mu )$ is ergodic if every G-invariant Borel set $A\subset X$ satisfies $\mu (A)=0$ or $\mu (X\setminus A)=0$ . A non-singular action $G\curvearrowright (X,\mu )$ is called weakly mixing if for any ergodic pmp action $G\curvearrowright (Y,\nu )$ the diagonal product action $G\curvearrowright X\times Y$ is ergodic. If G is not compact and $G\curvearrowright (X,\mu )$ is pmp, we say that $G\curvearrowright X$ is mixing if

$$ \begin{align*} \lim_{g\rightarrow \infty}\mu(g\cdot A\cap B)=\mu(A)\mu(B)\quad\text{for every pair of Borel subsets } A,B\subset X. \end{align*} $$

Suppose that $G\curvearrowright (X,\mu )$ is a non-singular action and that $\mu $ is a probability measure. A sequence of Borel subsets $A_n\subset X$ is called almost invariant if

$$ \begin{align*} \sup_{g\in K}\mu(g\cdot A_n\triangle A_n)\rightarrow 0\quad\text{for every compact subset } K\subset G. \end{align*} $$

The action $G\curvearrowright (X,\mu )$ is called strongly ergodic if every almost invariant sequence $A_n\subset X$ is trivial, that is, $\mu (A_n)(1-\mu (A_n))\rightarrow 0$ . The strong ergodicity of $G\curvearrowright (X,\mu )$ only depends on the measure class of $\mu $ . When $(Y,\nu )$ is a standard measure space and $\nu $ is infinite, a non-singular action $G\curvearrowright (Y,\nu )$ is called strongly ergodic if $G\curvearrowright (Y,\nu ')$ is strongly ergodic, where $\nu '$ is a probability measure that is equivalent to $\nu $ .

Following [AIM19, Definition A.16], we say that a non-singular action $G\curvearrowright (X,\mu )$ is dissipative up to compact stabilizers if each ergodic component is of the form ${G\curvearrowright G/ K}$ , for a compact subgroup $K\subset G$ . By [AIM19, Theorem A.29] a non-singular action ${G\curvearrowright (X,\mu )}$ , with $\mu (X)=1$ , is dissipative up to compact stabilizers if and only if

$$ \begin{align*} \int_G \frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)<+\infty \quad\text{for a.e. } x\in X, \end{align*} $$

where $\unicode{x3bb} $ denotes the left invariant Haar measure on G. We say that $G\curvearrowright (X,\mu )$ is infinitely recurrent if for every non-negligible subset $A\subset X$ and every compact subset $K\subset G$ there exists $g\in G\setminus K$ such that $\mu (g\cdot A\cap A)>0$ . By [AIM19, Proposition A.28] and Lemma 2.1 below, a non-singular action $G\curvearrowright (X,\mu )$ , with $\mu (X)=1$ , is infinitely recurrent if and only if

$$ \begin{align*} \int_G \frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)=+\infty \quad\text{for a.e. } x\in X. \end{align*} $$

A non-singular action $G\curvearrowright (X,\mu )$ is called dissipative if it is essentially free and dissipative up to compact stabilizers. In that case there exists a standard measure space $(X_0,\mu _0)$ such that $G\curvearrowright X$ is conjugate with the action $G\curvearrowright G\times X_0: \;g\cdot (h,x)=(gh,x)$ . A non-singular action $G\curvearrowright (X,\mu )$ decomposes, uniquely up to a null set, as ${G\curvearrowright D\sqcup C}$ , where $G\curvearrowright D$ is dissipative up to compact stabilizers and $G\curvearrowright C$ is infinitely recurrent. When G is a countable group and $G\curvearrowright (X,\mu )$ is essentially free, we say that $G\curvearrowright X$ is conservative if it is infinitely recurrent.

Lemma 2.1. Suppose that G is an lcsc group with left invariant Haar measure $\unicode{x3bb} $ and that $(X,\mu )$ is a standard probability space. Assume that $G\curvearrowright (X,\mu )$ is a non-singular action that is infinitely recurrent. Then we have that

$$ \begin{align*} \int_G \frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)=+\infty \quad\text{for a.e. } x\in X. \end{align*} $$

Proof. Note that the set

$$ \begin{align*} D=\left\{x\in X:\int_G\frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)<+\infty\right\} \end{align*} $$

is G-invariant. Therefore, it suffices to show that $G\curvearrowright X$ is not infinitely recurrent under the assumption that D has full measure.

Let $\pi \colon (X,\mu )\rightarrow (Y,\nu )$ be the projection onto the space of ergodic components of $G\curvearrowright X$ . Then there exist a conull Borel subset $Y_0\subset Y$ and a Borel map $\theta \colon Y_0\rightarrow X$ such that $(\pi \circ \theta )(y)=y$ for every $y\in Y_0$ .

Write $X_y=\pi ^{-1}(\{y\})$ . By [AIM19, Theorem A.29], for a.e. $y\in Y$ there exists a compact subgroup $K_y\subset G$ such that $G\curvearrowright X_y$ is conjugate with $G\curvearrowright G/ K_y$ . Let $G_n\subset G$ be an increasing sequence of compact subsets of G such that $\bigcup _{n\geq 1}\overset {\circ }{G}_n=G$ . For every $x\in X$ , write $G_x=\{g\in G:g\cdot x=x\}$ for the stabilizer subgroup of x. Using an argument as in [MRV11, Lemma 10], one shows that for each $n\geq 1$ the set $\{x\in X:G_x\subset G_n\}$ is Borel. Thus, for every $n\geq 1$ the set

$$ \begin{align*} U_n=\{y\in Y_0:K_y\subset G_n\}=\{y\in Y_0:G_{\theta(y)}\subset G_n\} \end{align*} $$

is a Borel subset of Y and we have that $\nu (\bigcup _{n\geq 1 } U_n)=1$ . Therefore, the sets

$$ \begin{align*} A_n=\{g\cdot \theta(y):g\in G_n, y\in U_{n}\} \end{align*} $$

are analytic and exhaust X up to a set of measure zero. So there exist an $n_0\in \mathbb {N}$ and a non-negligible Borel set $B\subset A_{n_0}$ . Suppose that $h\in G$ is such that $h\cdot B\cap B\neq \emptyset $ . Then there exist $y\in U_{n_0}$ and $g_1,g_2\in G_{n_0}$ such that $hg_1\cdot \theta (y)=g_2\cdot \theta (y)$ , and we get that $h\in G_{n_0}K_yG_{n_0}^{-1}\subset G_{n_0}G_{n_0}G_{n_0}^{-1}$ . In other words, for $h\in G$ outside the compact set $G_{n_0}G_{n_0}G_{n_0}^{-1}$ we have that $\mu (h\cdot B \cap B)=0$ , so that $G\curvearrowright X$ is not infinitely recurrent.

We will frequently use the following result of Schmidt and Walters. Suppose that ${G\curvearrowright (X,\mu )}$ is a non-singular action that is infinitely recurrent and suppose that ${G\curvearrowright (Y,\nu )}$ is pmp and mixing. Then by [SW81, Theorem 2.3] we have that

$$ \begin{align*} L^{\infty}(X\times Y)^{G}=L^{\infty}(X)^{G}\mathbin{\overline{\otimes}} 1, \end{align*} $$

where $G\curvearrowright X\times Y$ acts diagonally. Although [SW81, Theorem 2.3] demands proper ergodicity of the action $G\curvearrowright (X,\mu )$ , the infinite recurrence assumption is sufficient as remarked in [AIM19, Remark 7.4].

2.2 The Maharam extension and crossed products

Let $(X,\mu )$ be a standard measure space. For any non-singular automorphism $\varphi \in \operatorname {Aut}(X,\mu )$ , we define its Maharam extension by

$$ \begin{align*} \widetilde{\varphi}\colon X\times \mathbb{R}\rightarrow X\times \mathbb{R}:\quad \widetilde{\varphi}(x,t)=(\varphi(x),t+\log(d\varphi^{-1}\mu/d\mu) (x)). \end{align*} $$

Then $\widetilde {\varphi }$ preserves the infinite measure $\mu \times \exp (-t)dt$ . The assignment $\varphi \mapsto \widetilde {\varphi }$ is a continuous group homomorphism from $\operatorname {Aut}(X)$ to $\operatorname {Aut}(X\times \mathbb {R})$ . Thus, for each non-singular group action $G\curvearrowright (X,\mu )$ , by composing with this map, we obtain a non-singular group action $G\curvearrowright X\times \mathbb {R}$ , which we call the Maharam extension of $G\curvearrowright X$ . If $G\curvearrowright X$ is a non-singular group action, the translation action $\mathbb {R}\curvearrowright X\times \mathbb {R}$ in the second component commutes with the Maharam extension $G\curvearrowright X\times \mathbb {R}$ . Therefore, we get a well-defined action $\mathbb {R}\curvearrowright L^{\infty }(X\times \mathbb {R})^{G}$ , which is the Krieger flow associated to the action $G\curvearrowright X$ . The Krieger flow is given by $\mathbb {R}\curvearrowright \mathbb {R}$ if and only if there exists a G-invariant $\sigma $ -finite measure $\nu $ on X that is equivalent to $\mu $ .

Suppose that $M\subset B(\mathcal {H})$ is a von Neumann algebra represented on the Hilbert space $\mathcal {H}$ and that $\alpha \colon G\curvearrowright M$ is a continuous action on M of an lcsc group G. Then the crossed product von Neumann algebra $M\rtimes _{\alpha } G\subset B(L^2(G,\mathcal {H}))$ is the von Neumann algebra generated by the operators $\{\pi (x)\}_{x\in M}$ and $\{u_h\}_{h\in G}$ acting on $\xi \in L^2(G,\mathcal {H})$ as

$$ \begin{align*} (\pi(x)\xi)(g)=\alpha_{g^{-1}}(x)\xi(g),\quad(u_h\xi)(g)=\xi(h^{-1}g). \end{align*} $$

In particular, if $G\curvearrowright (X,\mu )$ is a non-singular group action, the crossed product $L^{\infty }(X)\rtimes G\subset B(L^2(G\times X))$ is the von Neumann algebra generated by the operators

$$ \begin{align*} (\pi(H)\xi)(g,x)=H(g\cdot x)\xi(g,x), \quad (u_h\xi)(g,x)=\xi(h^{-1}g,x), \end{align*} $$

for $H\in L^{\infty }(X)$ and $h\in G$ . If $G\curvearrowright X$ is non-singular essentially free and ergodic, then $L^{\infty }(X)\rtimes G$ is a factor. Moreover, when G is a unimodular group, the Krieger flow of ${G\curvearrowright X}$ equals the flow of weights of the crossed product von Neumann algebra $L^{\infty }(X)\rtimes G$ . For non-unimodular groups this is not necessarily true, motivating the following definition.

Definition 2.2. Let G be an lcsc group with modular function $\Delta \colon G\rightarrow \mathbb {R}_{>0}$ . Let $\unicode{x3bb} $ denote the Lebesgue measure on $\mathbb {R}$ . Suppose that $\alpha \colon G\curvearrowright (X,\mu )$ is a non-singular action. We define the modular Maharam extension of $G\curvearrowright X$ as the non-singular action

$$ \begin{align*} \beta\colon G\curvearrowright (X\times \mathbb{R}, \mu\times \unicode{x3bb}): \quad g\cdot(x,t)=(g\cdot x, t+\log(\Delta(g))+\log(dg^{-1}\mu/d\mu)(x)). \end{align*} $$

Let $L^{\infty }(X\times \mathbb {R})^{\beta }$ denote the subalgebra of $\beta $ -invariant elements. We define the flow of weights associated to $G{\kern-1pt}\curvearrowright{\kern-1pt} X$ as the translation action $\mathbb {R}{\kern-1pt}\curvearrowright{\kern-1pt} L^{\infty }(X{\kern-1pt}\times{\kern-1pt} \mathbb {R})^{\beta }: (t\cdot H)(x,s)= H(x,s-t)$ .

As we explain below, the flow of weights associated to an essentially free ergodic non-singular action $G\curvearrowright X$ equals the flow of weights of the crossed product factor $L^{\infty }(X)\rtimes G$ , justifying the terminology. See also [Sa74, Proposition 4.1].

Let $\alpha \colon G\curvearrowright X$ be an essentially free ergodic non-singular group action with modular Maharam extension $\beta \colon G\curvearrowright X\times \mathbb {R}$ . By [Sa74, Proposition 1.1] there is a canonical normal semifinite faithful weight $\varphi $ on $L^{\infty }(X)\rtimes _{\alpha } G$ such that the modular automorphism group $\sigma ^{\varphi }$ is given by

$$ \begin{align*} \sigma^{\varphi}_t(\pi(H))=\pi(H), \quad \sigma^{\varphi}_t(u_g)=\Delta(g)^{it}u_g\pi((dg^{-1}\mu/d\mu)^{it}), \end{align*} $$

where $\Delta \colon G\rightarrow \mathbb {R}_{>0}$ denotes the modular function of G.

For an element $\xi \in L^2(\mathbb {R}, L^2(G\times X))$ and $(g,x)\in G\times X$ , write $\xi _{g,x}$ for the map given by $\xi _{g,x}(s)=\xi (s,g,x)$ . Then by Fubini’s theorem $\xi _{g,x}\in L^2(\mathbb {R})$ for a.e. ${(g,x)\in G\times X}$ . Let $U\colon L^2(\mathbb {R}, L^2(G\times X))\rightarrow L^2(G,L^2(X\times \mathbb {R}))$ be the unitary given on ${\xi \in L^2(\mathbb {R}, L^2(G\times X))}$ by

$$ \begin{align*} (U\xi)(g,x,t)=\mathcal{F}^{-1}(\xi_{g,x})(t+\log(\Delta(g))+\log(dg^{-1}\mu/d\mu)(x)), \end{align*} $$

where $\mathcal {F}^{-1}\colon L^2(\mathbb {R})\rightarrow L^2(\mathbb {R})$ denotes the inverse Fourier transform. One can check that conjugation by U induces an isomorphism

$$ \begin{align*} \Psi\colon (L^{\infty}(X)\rtimes_{\alpha} G)\rtimes_{\sigma^{\varphi}}\mathbb{R}\rightarrow L^{\infty}(X\times \mathbb{R})\rtimes_{\beta}G. \end{align*} $$

Let $\kappa \colon L^{\infty }(X\times \mathbb {R})\rightarrow L^{\infty }(X\times \mathbb {R})\rtimes _{\beta }G$ be the inclusion map and let $\gamma \colon \mathbb {R}\curvearrowright L^{\infty }(X\times \mathbb {R})\rtimes _{\beta }G$ be the action given by

$$ \begin{align*} \gamma_t(\kappa(H))(x,s)=\kappa(H)(x,s-t),\quad\gamma_t(u_g)=u_g. \end{align*} $$

Then one can verify that $\Psi $ conjugates the dual action $\widehat {\sigma ^{\varphi }}\colon \mathbb {R}\curvearrowright (L^{\infty }(X)\rtimes _{\alpha } G)\rtimes _{\sigma ^{\varphi }}\mathbb {R}$ and $\gamma $ . Therefore, we can identify the flow of weights $\mathbb {R}\curvearrowright \mathcal {Z}((L^{\infty }(X)\rtimes _{\alpha } G)\rtimes _{\sigma ^{\varphi }}\mathbb {R})$ with $\mathbb {R}\curvearrowright \mathcal {Z}(L^{\infty }(X\times \mathbb {R})\rtimes _{\beta } G)\cong L^{\infty }(X\times \mathbb {R})^{\beta }$ : the flow of weights associated to ${G\curvearrowright X}$ .

Remark 2.3. It will be useful to speak about the Krieger type of a non-singular ergodic action $G\curvearrowright X$ . In light of the discussion above, we will only use this terminology for countable groups G, so that no confusion arises with the type of the crossed product von Neumann algebra $L^{\infty }(X)\rtimes G$ . So assume that G is countable and that $G\curvearrowright (X,\mu )$ is a non-singular ergodic action. Then the Krieger flow is ergodic and we distinguish several cases. If $\nu $ is atomic, we say that $G\curvearrowright X$ is of type I. If $\nu $ is non-atomic and finite, we say that $G\curvearrowright X$ is of type II $_{1}$ . If $\nu $ is non-atomic and infinite, we say that $G\curvearrowright X$ is of type II $_{\infty }$ . If the Krieger flow is given by $\mathbb {R}\curvearrowright \mathbb {R}/\log (\unicode{x3bb} )\mathbb {Z}$ with $\unicode{x3bb} \in (0,1)$ , we say that $G\curvearrowright X$ is of type III $_{\unicode{x3bb} }$ . If the Krieger flow is the trivial flow $\mathbb {R}\curvearrowright \{\ast \}$ , we say that $G\curvearrowright X$ is of type III $_{1}$ . If the Krieger flow is properly ergodic (that is, every orbit has measure zero), we say that $G\curvearrowright X$ is of type III $_{0}$ .

2.3 Non-singular Bernoulli actions

Suppose that G is a countable infinite group and that $(\mu _g)_{g\in G}$ is a family of equivalent probability measures on a standard Borel space $X_0$ . The action

(2.1) $$ \begin{align} G\curvearrowright (X,\mu)=\prod_{h\in G}(X_0,\mu_h):\quad (g\cdot x)_h=x_{g^{-1}h} \end{align} $$

is called the Bernoulli action. For two probability measures $\nu ,\eta $ on a standard Borel space Y, the Hellinger distance $H^2(\nu ,\eta )$ is defined by

(2.2) $$ \begin{align} H^2(\nu,\eta)=\frac{1}{2}\int_Y\left(\sqrt{d\nu/d\zeta}-\sqrt{d\eta/d\zeta}\right)^2d\zeta, \end{align} $$

where $\zeta $ is any probability measure on Y such that $\nu ,\eta \prec \zeta $ . By Kakutani’s criterion for equivalence of infinite product measures [Kak48] the Bernoulli action (2.1) is non-singular if and only if

$$ \begin{align*} \sum_{h\in G}H^2(\mu_{h},\mu_{gh})<+\infty \quad\text{for every } g\in G. \end{align*} $$

If $(X,\mu )$ is non-atomic and the Bernoulli action (2.1) is non-singular, then it is essentially free by [BKV19, Lemma 2.2].

Suppose that I is a countable infinite set and that $(\mu _i)_{i\in I}$ is a family of equivalent probability measures on a standard Borel space $X_0$ . If G is an lcsc group that acts on I, the action

(2.3) $$ \begin{align} G\curvearrowright (X,\mu)=\prod_{i\in I}(X_0,\mu_i):\quad (g\cdot x)_i=x_{g^{-1}\cdot i} \end{align} $$

is called the generalized Bernoulli action and it is non-singular if and only if $\sum _{i\in I}H^2(\mu _i,\mu _{g\cdot i})<+\infty $ for every $g\in G$ . When $\nu $ is a probability measure on $X_0$ such that $\mu _i=\nu $ for every $i\in I$ , the generalized Bernoulli action (2.3) is pmp and it is mixing if and only if the stabilizer subgroup $G_i=\{g\in G:g\cdot i=i\}$ is compact for every $i\in I$ . In particular, if G is countable infinite, the pmp Bernoulli action $G\curvearrowright (X_0,\mu _0)^{G}$ is mixing.

2.4 Groups acting on trees

Let $T=(V(T),E(T))$ be a locally finite tree, so that the edge set $E(T)$ is a symmetric subset of $ V(T)\times V(T)$ with the property that vertices $v,w\in V(T)$ are adjacent if and only if $(v,w),(w,v)\in E(T)$ . When T is clear from the context, we will write E instead of $E(T)$ . Also we will often write T instead of $V(T)$ for the vertex set. For any two vertices $v,w\in T$ let $[v,w]$ denote the smallest subtree of T that contains v and w. The distance between vertices $v,w\in T$ is defined as ${d(v,w)=|V([v,w])|-1}$ . Fixing a root $\rho \in T$ , we define the boundary $\partial T$ of T as the collection of all infinite line segments starting at $\rho $ . We equip $\partial T$ with a metric $d_\rho $ as follows. If $\omega ,\omega '\in \partial T$ , let $v\in T$ be the unique vertex such that $d(\rho ,v)=\sup _{v\in \omega \cap \omega '}d(\rho , v)$ and define

$$ \begin{align*} d_\rho(\omega,\omega')=\exp(-d(\rho,v)). \end{align*} $$

Then, up to homeomorphism, the space $(\partial T, d_{\rho })$ does not depend on the chosen root $\rho \in T$ . Furthermore, the Hausdorff dimension $\dim _H \partial T$ of $(\partial T, d_\rho )$ is also independent of the choice of $\rho \in T$ .

Let $\operatorname {Aut}(T)$ denote the group of automorphisms of T. By [Tit70, Proposition 3.2], if $g\in \operatorname {Aut}(T)$ , then either:

• • g fixes a vertex or interchanges a pair of vertices (in this case we say that g is elliptic);

• • or there exists a bi-infinite line segment $L\subset T$ , called the axis of g, such that g acts on L by non-trivial translation (in this case we say that g is hyperbolic).

We equip $\operatorname {Aut}(T)$ with the topology of pointwise convergence. A subgroup $G\subset \operatorname {Aut}(T)$ is closed with respect to this topology if and only if for every $v\in T$ the stabilizer subgroup $G_v=\{g\in G:g\cdot v= v\}$ is compact. An action of an lcsc group G on T is a continuous homomorphism $G\rightarrow \operatorname {Aut}(T)$ . We say that the action $G\curvearrowright T$ is cocompact if there is a finite set $F\subset E(T)$ such that $G\cdot F=E(T)$ . A subgroup $G\subset \operatorname {Aut}(T)$ is called non-elementary if it does not fix any point in $T\cup \partial T$ and does not interchange any pair of points in $T\cup \partial T$ . Equivalently, $G\subset \operatorname {Aut}(T)$ is non-elementary if there exist hyperbolic elements $h,g\in G$ with axes $L_h$ and $L_g$ such that $L_h\cap L_g$ is finite. If $G\subset \operatorname {Aut}(T)$ is a non-elementary closed subgroup, there exists a unique minimal G-invariant subtree $S\subset T$ and G is compactly generated if and only if $G\curvearrowright S$ is cocompact (see [CM11, §2]). Recall from (1.5) the definition of the Poincaré exponent $\delta (G\curvearrowright T)$ of a subgroup $G\subset \operatorname {Aut}(T)$ . If $G\subset \operatorname {Aut}(T)$ is a closed subgroup such that $G\curvearrowright T$ is cocompact, then we have that $\delta (G\curvearrowright T)=\dim _{H}\partial T$ .

3 Phase transitions of non-singular Bernoulli actions: proof of Theorems A and B

Let G be a countable infinite group and let $(\mu _g)_{g\in G}$ be a family of equivalent probability measures on a standard Borel space $X_0$ . Let $\nu $ also be a probability measure on $X_0$ . For $t\in [0,1]$ we define the family of probability measures

(3.1) $$ \begin{align} \mu_g^t=(1-t)\nu+t\mu_g, \quad g\in G. \end{align} $$

We write $\mu _t$ for the infinite product measure $\mu _t=\prod _{g\in G}\mu _g^t$ on $X=\prod _{g\in G}X_0$ . We prove Theorem 3.1 below, which is slightly more general than Theorem A.

Theorem 3.1. Let G be a countable infinite group and let $(\mu _g)_{g\in G}$ be a family of equivalent probability measures on a standard probability space $X_0$ , which is not supported on a single atom. Assume that the Bernoulli action $G\curvearrowright \prod _{g\in G}(X_0,\mu _g)$ is non-singular. Let $\nu $ also be a probability measure on $X_0$ . Then for every $t\in [0,1]$ the Bernoulli action

(3.2) $$ \begin{align} G\curvearrowright (X,\mu_t)=\prod_{g\in G}(X_0,(1-t)\nu+t\mu_g) \end{align} $$

is non-singular. Assume, in addition, that one of the following conditions holds.

1. (1) $\nu \sim \mu _e$ .

2. (2) $\nu \prec \mu _e $ and $\sup _{g\in G}|{\log}\ d\mu _g/d\mu _e(x)|<+\infty $ for a.e $x\in X_0$ .

Then there exists a $t_1\in [0,1]$ such that $G\curvearrowright (X,\mu _t)$ is dissipative for every $t>t_1$ and weakly mixing for every $t<t_1$ .

Remark 3.2. One might hope to prove a completely general phase transition result that only requires $\nu \prec \mu _e$ , and not the additional assumption that $\sup _{g\in G}|{\log}\ d\mu _g/d\mu _e(x)|<+\infty $ for a.e. $x\in X_0$ . However, the following example shows that this is not possible.

Let G be any countable infinite group and let $G\curvearrowright \prod _{g\in G}(C_0,\eta _g)$ be a conservative non-singular Bernoulli action. Note that Theorem 3.1 implies that

$$ \begin{align*}G\curvearrowright \prod_{g\in G}(C_0,(1-t)\eta_e+t\eta_g)\end{align*} $$

is conservative for every $t<1$ . Let $C_1$ be a standard Borel space and let $(\mu _g)_{g\in G}$ be a family of equivalent probability measures on $X_0\kern1.2pt{=}\kern1.2pt C_0\kern1.2pt{\sqcup}\kern1.2pt C_1$ such that ${0\kern1.2pt{<}\kern1pt\sum _{g\in G}\mu _g(C_1)\kern1.2pt{<}\kern1.2pt{+}\kern0.5pt\infty }$ and such that $\mu _g |_{C_0}\kern1.2pt{=}\kern1.2pt\mu _g(C_0)\eta _g$ . Then the Bernoulli action $G\kern1.2pt{\curvearrowright}\kern1.2pt (X,\mu )\kern1.2pt{=}\kern1.2pt\prod _{g\in G}(X_0,\mu _g)$ is non-singular with non-negligible conservative part $C_0^{G}\subset G$ and dissipative part $X\setminus C_0^G$ . Taking $\nu =\eta _e\prec \mu _e$ , for each $t<1$ the Bernoulli action $G\curvearrowright (X,\mu _t)= \prod _{g\in G}(X_0,(1-t)\eta _e+t\mu _g)$ is constructed in the same way, by starting with the conservative Bernoulli action $G\curvearrowright \prod _{g\in G}(C_0,(1-t)\eta _e+t\eta _g)$ . So for every $t\in (0,1)$ the Bernoulli action $G\curvearrowright (X,\mu _t)$ has non-negligible conservative part and non-negligible dissipative part.

We can also prove a version of Theorem B in the more general setting of Theorem 3.1.

Theorem 3.3. Let G be a countable infinite non-amenable group. Make the same assumptions as in Theorem 3.1 and consider the non-singular Bernoulli actions ${G\curvearrowright (X,\mu _t)}$ given by (3.2). Assume, moreover, that:

1. (1) $\nu \sim \mu _e$ , or

2. (2) $\nu \prec \mu _e$ and $\sup _{g\in G}|{\log}\ d\mu _g/d\mu _e(x)|<+\infty $ for a.e. $x\in X_0$ .

Then there exists a $t_0>0$ such that $G\curvearrowright (X,\mu _t)$ is strongly ergodic for every $t<t_0$ .

Proof of Theorem 3.1

Assume that $G\curvearrowright (X,\mu _1)=\prod _{g\in G}(X_0,\mu _g)$ is non-singular. For every $t\in [0,1]$ we have that

$$ \begin{align*} \sum_{h\in G}H^2(\mu_h^t,\mu_{gh}^t)\leq t\sum_{h\in G}H^2(\mu_h, \mu_{gh}) \quad \text{for every } g\in G, \end{align*} $$

so that $G\curvearrowright (X,\mu _t)$ is non-singular for every $t\in [0,1]$ . The rest of the proof we divide into two steps.

Claim 1. If $G\curvearrowright (X,\mu _t)$ is conservative, then $G\curvearrowright (X,\mu _s)$ is weakly mixing for every $s<t$ .

Proof of Claim 1

Note that for every $g\in G$ we have that

$$ \begin{align*} (\mu_g^s)^r&=(1-r)\nu+r\mu_g^s=(1-r)\nu+r(1-s)\nu+rs\mu_g=\mu_g^{sr}, \end{align*} $$

so that $(\mu _s)_r=\mu _{sr}$ . Therefore, it suffices to prove that $G\curvearrowright (X,\mu _s)$ is weakly mixing for every $s<1$ , assuming that $G\curvearrowright (X,\mu _1)$ is conservative.

The claim is trivially true for $s=0$ . So assume that $G\curvearrowright (X,\mu _1)$ is conservative and fix $s\in (0,1)$ . Let $G\curvearrowright (Y,\eta )$ be an ergodic pmp action. Define $Y_0=X_0\times X_0\times \{0,1\}$ and define the probability measures $\unicode{x3bb} $ on $\{0,1\}$ by $\unicode{x3bb} (0)=s$ . Define the map $\theta \colon Y_0\rightarrow X_0$ by

(3.3) $$ \begin{align} \theta(x,x',j)=\begin{cases}x &\text{if } j=0,\\ x' &\text{if } j=1.\end{cases} \end{align} $$

Then for every $g\in G$ we have that $\theta _*(\mu _g\times \nu \times \unicode{x3bb} )=\mu _g^s$ . Write $Z=\{0,1\}^G$ and equip Z with the probability measure $\unicode{x3bb} ^{G}$ . We identify the Bernoulli action $G\curvearrowright Y_0^{G}$ with the diagonal action $G\curvearrowright X\times X\times Z$ . By applying $\theta $ in each coordinate we obtain a G-equivariant factor map

(3.4) $$ \begin{align} \Psi\colon X\times X\times Z\rightarrow X:\quad \Psi(x,x',z)_h=\theta(x_h,x^{\prime}_h,z_h). \end{align} $$

Then the map $\mathord {\textrm {id}}_Y\times \Psi \colon Y\times X\times X\times Z\rightarrow Y\times X$ is G-equivariant and we have that $(\mathord {\textrm { id}}_Y\times \Psi )_*(\eta \times \mu _1\times \mu _0\times \unicode{x3bb} ^G)=\eta \times \mu _s$ . The construction above is similar to [KS20, §4].

Take $F\in L^{\infty }(Y\times X,\eta \times \mu _s)^{G}$ . Note that the diagonal action $G\curvearrowright (Y\times X,\eta \times \mu _1)$ is conservative, since $G\curvearrowright (Y,\eta )$ is pmp. The action $G\curvearrowright (X\times Z,\mu _0\times \unicode{x3bb} ^{G})$ can be identified with a pmp Bernoulli action with base space $(X_0\times \{0,1\},\nu \times \unicode{x3bb} )$ , so that it is mixing. By [SW81, Theorem 2.3] we have that

$$ \begin{align*} L^{\infty}(Y\times X\times X\times Z,\eta\times \mu_1\times \mu_0\times \unicode{x3bb}^{G})^{G}=L^{\infty}(Y\times X,\eta\times \mu_1)^{G}\mathbin{\overline{\otimes}} 1\mathbin{\overline{\otimes}} 1, \end{align*} $$

which implies that the assignment $(y,x,x',z)\mapsto F(y, \Psi (x,x',z))$ is essentially independent of $x'$ and z. Choosing a finite set of coordinates $\mathcal {F}\subset G$ and changing, for $g\in \mathcal {F}$ , the value $z_g$ between $0$ and $1$ , we see that F is essentially independent of the $x_g$ -coordinates for $g\in \mathcal {F}$ . As this is true for any finite set $\mathcal {F}\subset G$ , we have that $F\in L^{\infty }(Y)^{G}\mathbin {\overline {\otimes }} 1$ . The action $G\curvearrowright (Y,\eta )$ is ergodic and therefore F is essentially constant. We conclude that $G\curvearrowright (X,\mu _s)$ is weakly mixing.

Claim 2. If $\nu \sim \mu _e$ and if $G\curvearrowright (X,\mu _t)$ is not dissipative, then $G\curvearrowright (X,\mu _s)$ is conservative for every $s<t$ .

Proof of Claim 2

Again it suffices to assume that $G\curvearrowright (X,\mu _1)$ is not dissipative and to show that $G\curvearrowright (X,\mu _s)$ is conservative for every $s<1$ .

When $s=0$ , the statement is trivial, so assume that $G\curvearrowright (X,\mu _1)$ is not dissipative and fix $s\in (0,1)$ . Let $C\subset X$ denote the non-negligible conservative part of $G\curvearrowright (X,\mu _1)$ . As in the proof of Claim 1, write $Z=\{0,1\}^{G}$ and let $\unicode{x3bb} $ be the probability measure on $\{0,1\}$ given by $\unicode{x3bb} (0)=s$ . Writing $\Psi \colon X\times X\times Z\rightarrow X$ for the G-equivariant map (3.4). We claim that $\Psi _*((\mu _1\times \mu _0\times \unicode{x3bb} ^{G})|_{C\times X\times Z})\sim \mu _s$ , so that $G\curvearrowright (X,\mu _s)$ is a factor of a conservative non-singular action, and therefore must be conservative itself.

As $\Psi _*(\mu _1\times \mu _0\times \unicode{x3bb} ^{G})=\mu _s$ , we have that $\Psi _*((\mu _1\times \mu _0\times \unicode{x3bb} ^{G}) |_{C\times X\times Z})\prec \mu _s$ . Let $\mathcal {U}\subset X$ be the Borel set, uniquely determined up to a set of measure zero, such that $\Psi _*((\mu _1\times \mu _0\times \unicode{x3bb} ^{G}) |_{C\times X\times Z})\sim \mu _s |_{\mathcal {U}}$ . We have to show that $\mu _s(X\setminus \mathcal {U})=0$ . Fix a finite subset $\mathcal {F}\subset G$ . For every $t\in [0,1]$ define

$$ \begin{align*} (X_1,\gamma_1^{t})&=\prod_{g\in \mathcal{F}}(X_0,(1-t)\nu+t\mu_g),\\ (X_2,\gamma_2^{t})&=\prod_{g\in G \setminus \mathcal{F}}(X_0,(1-t)\nu+t\mu_g). \end{align*} $$

We shall write $\gamma _1=\gamma _1^1, \gamma _2=\gamma _2^1$ . Also define

$$ \begin{align*} (Y_1,\zeta_1)&=\prod_{g\in \mathcal{F}}(X_0\times X_0\times \{0,1\},\mu_g\times \nu\times \unicode{x3bb}),\\ (Y_2,\zeta_2)&=\prod_{g\in G\setminus \mathcal{F}}(X_0\times X_0\times \{0,1\},\mu_g\times \nu\times \unicode{x3bb}). \end{align*} $$

By applying the map (3.3) in every coordinate, we get factor maps $\Psi _j\colon Y_j\rightarrow X_j$ that satisfy $(\Psi _j)_*(\zeta _j)=\gamma _j^{s}$ for $j=1,2$ . Identify $X_1\times Y_2\cong X\times (X_0\times \{0,1\})^{G\setminus \mathcal {F}}$ and define the subset $C'\subset X_1\times Y_2$ by $C'=C\times (X_0\times \{0,1\})^{G\setminus \mathcal {F}}$ . Let $\mathcal {U}'\subset X$ be Borel such that

$$ \begin{align*} (\mathord{\textrm{id}}_{X_1}\times \Psi_2)_*((\gamma_1\times \zeta_2)|_{C'})\sim (\gamma_1\times \gamma_2^{s})|_{\mathcal{U}'}. \end{align*} $$

Identify $Y_1\times X_2\cong X\times (X_0\times \{0,1\})^{\mathcal {F}}$ and define $V\subset Y_1\times X_2$ by $V=\mathcal {U}'\times (X_0\times \{0,1\})^{\mathcal {F}}$ . Then we have that

$$ \begin{align*} (\Psi_1\times \mathord{\textrm{id}}_{X_2})_*((\zeta_1\times \gamma_2^s)|_{V})&\sim (\Psi_1\times \mathord{\textrm{ id}}_{X_2})_*(\mathord{\textrm{id}}_{Y_1}\times \Psi_2)_*((\gamma_1\times \zeta_1)|_{C'}\times \nu^{\mathcal{F}}\times \unicode{x3bb}^{\mathcal{F}})\\ &=\Psi_*((\zeta_1\times \zeta_2)|_{C\times X\times Z})\sim \mu_s|_{\mathcal{U}}. \end{align*} $$

Let $\pi \colon X_1\times X_2\rightarrow X_2$ and $\pi '\colon Y_1\times X_2\rightarrow X_2$ denote the coordinate projections. Note that by construction we have that

(3.5) $$ \begin{align} \pi^{\prime}_*((\zeta_1\times \gamma_2^s)|_V) \sim \pi_*((\gamma_1\times \gamma_2^{s})|_{\mathcal{U}'})\sim \pi_*(\mu_s|_{\mathcal{U}}). \end{align} $$

Let $W\subset X_2$ be Borel such that $\pi _*(\mu _s |_{\mathcal {U}})\sim \gamma _2^s |_{W}$ . For every $y\in X_2$ define the Borel sets

$$ \begin{align*} \mathcal{U}_y=\{x\in X_1:(x,y)\in \mathcal{U}\}\quad \text{and} \quad\mathcal{U}^{\prime}_y=\{x\in X_1:(x,y)\in \mathcal{U}'\}. \end{align*} $$

As $\pi _*((\gamma _1\times \gamma _2^s) |_{\mathcal {U}'})\sim \gamma _2^s |_{W}$ , we have that

$$ \begin{align*} \gamma_1(\mathcal{U}^{\prime}_y)>0 \quad\text{for } \gamma_2^s\text{-a.e. } y\in W. \end{align*} $$

The disintegration of $(\gamma _1\times \gamma _2^s) |_{\mathcal {U}'}$ along $\pi $ is given by $(\gamma _1 |_{\mathcal {U}^{\prime }_y})_{y\in W}$ . Therefore, the disintegration of $(\zeta _1\times \gamma _2^s) |_{V}$ along $\pi '$ is given by $(\gamma _1 |_{\mathcal {U}^{\prime }_y}\times \nu ^{\mathcal {F}}\times \unicode{x3bb} ^{\mathcal {F}})_{y\in W}$ . We conclude that the disintegration of $(\Psi _1\times \mathord {\textrm {id}}_{X_2})_*((\zeta _1\times \gamma _2^s) |_V)$ along $\pi $ is given by $((\Psi _1)_*(\gamma _1 |_{\mathcal {U}^{\prime }_y}\times \nu ^{\mathcal {F}}\times \unicode{x3bb} ^{\mathcal {F}}))_{y\in W}$ . The disintegration of $\mu _s |_{\mathcal {U}}$ along $\pi $ is given by $(\gamma _2^s |_{\mathcal {U}_y})_{y\in W}$ . Since $\mu _s |_{\mathcal {U}}\sim (\Psi _1\times \mathord {\textrm { id}}_{X_2})_*((\zeta _1\times \gamma _2^s) |_V)$ , we conclude that

$$ \begin{align*} (\Psi_1)_*(\gamma_1|_{\mathcal{U}^{\prime}_y}\times \nu^{\mathcal{F}}\times \unicode{x3bb}^{\mathcal{F}})\sim \gamma_1^s|_{\mathcal{U}_y}\quad\text{for } \gamma_2^s\text{-a.e. } y\in W. \end{align*} $$

As $\gamma _1(\mathcal {U}^{\prime }_y)>0$ for $\gamma _2^s$ -a.e. $y\in W$ , and using that $\nu \sim \mu _e$ , we see that

$$ \begin{align*} \gamma_1^{s}\sim \nu^{\mathcal{F}}&\sim(\Psi_1)_*((\gamma_1\times \nu^{\mathcal{F}}\times \unicode{x3bb}^{\mathcal{F}})|_{\mathcal{U}^{\prime}_y\times X_0^{\mathcal{F}}\times \{1\}^{\mathcal{F}}})\\ &\prec (\Psi_1)_*(\gamma_1|_{\mathcal{U}^{\prime}_y}\times \nu^{\mathcal{F}}\times \unicode{x3bb}^{\mathcal{F}}). \end{align*} $$

for $\gamma _2^{s}$ -a.e. $y\in W$ . It is clear that also $(\Psi _1)_*(\gamma _1 |_{\mathcal {U}^{\prime }_y}\times \nu ^{\mathcal {F}}\times \unicode{x3bb} ^{\mathcal {F}})\prec \gamma _1^{s}$ , so that $\gamma _1^{s} |_{\mathcal {U}_y}\sim \gamma _1^{s}$ for $\gamma _2^s$ -a.e. $y\in W$ . Therefore, we have that $\gamma _1^s(X_1\setminus \mathcal {U}_y)=0$ for $\gamma _2^s$ -a.e. $y\in W$ , so that

$$ \begin{align*} \mu_s(\mathcal{U}\triangle (X_0^{{\mathcal{F}}}\times W))=0. \end{align*} $$

Since this is true for every finite subset $\mathcal {F}\subset G$ , we conclude that $\mu _s(X\setminus \mathcal {U})=0$ .

The conclusion of the proof now follows by combining both claims. Assume that ${G\curvearrowright (X,\mu _t)}$ is not dissipative and fix $s<t$ . Choose r such that $s<r<t$ .

$\nu \sim \mu _e$ . By Claim 2 we have that $G\curvearrowright (X,\mu _r)$ is conservative. Then by Claim 1 we see that $G\curvearrowright (X,\mu _s)$ is weakly mixing.

$\nu \prec \mu _e$ . As $\nu \prec \mu _e$ , the measures $\mu _e^{t}$ and $\mu _e$ are equivalent. We have that

$$ \begin{align*} \frac{d\mu_g^{t}}{d\mu_e^{t}}=\bigg((1-t)\frac{d\nu}{d\mu_e}+t\frac{d\mu_g}{d\mu_e}\bigg)\frac{d\mu_e}{d\mu_e^{t}}. \end{align*} $$

So if $\sup _{g\in G}|{\log}\ d\mu _g/d\mu _e(x)|<+\infty $ for a.e $x\in X_0$ , we also have that

$$ \begin{align*}\sup_{g\in G}|{\log}\ d\mu_g^{t}/d\mu_e^{t}(x)|<+\infty \quad\text{for a.e. }x\in X_0.\end{align*} $$

It follows from [BV20, Proposition 4.3] that $G\curvearrowright (X,\mu _t)$ is conservative. Then by Claim 1 we have that $G\curvearrowright (X,\mu _s)$ is weakly mixing.

Remark 3.4. Let I be a countably infinite set and suppose that we are given a family of equivalent probability measures $(\mu _i)_{i\in I}$ on a standard Borel space $X_0$ . Let $\nu $ be a probability measure on $X_0$ that is equivalent to all the $\mu _i$ . If G is an lcsc group that acts on I such that for each $i \in I$ the stabilizer subgroup $G_i=\{g\in G:g\cdot i=i\}$ is compact, then the pmp generalized Bernoulli action

$$ \begin{align*} G\curvearrowright \prod_{i\in I}(X_0,\nu), \quad (g\cdot x)_{i}=x_{g^{-1}\cdot i} \end{align*} $$

is mixing. For $t\in [0,1]$ write

$$ \begin{align*} (X,\mu_t)=\prod_{i \in I}(X_0,(1-t)\nu+t\mu_i) \end{align*} $$

and assume that the generalized Bernoulli action $G\curvearrowright (X,\mu _1)$ is non-singular.

Since [SW81, Theorem 2.3] still applies to infinitely recurrent actions of lcsc groups (see [AIM19, Remark 7.4]), it is straightforward to adapt the proof of Claim 1 in the proof of Theorem 3.1 to prove that if $G\curvearrowright (X,\mu _t)$ is infinitely recurrent, then $G\curvearrowright (X,\mu _s)$ is weakly mixing for every $s<t$ . Similarly, we can adapt the proof of Claim 2, using that a factor of an infinitely recurrent action is again infinitely recurrent. Together, this leads to the following phase transition result in the lcsc setting.

Assume that $G_i=\{g\in G:g\cdot i=i\}$ is compact for every $i\in I$ and that $\nu \sim \mu _e$ . Then there exists a $t_1\in [0,1]$ such that $G\curvearrowright (X,\mu _t)$ is dissipative up to compact stabilizers for every $t>t_1$ and weakly mixing for every $t<t_1$ .

Recall the following definition from [BKV19, Definition 4.2]. When G is a countable infinite group and $G\curvearrowright (X,\mu )$ is a non-singular action on a standard probability space, a sequence $(\eta _n)$ of probability measures on G is called strongly recurrent for the action $G\curvearrowright (X,\mu )$ if

$$ \begin{align*} \sum_{h\in G}\eta_n^2(h)\int_{X}\frac{d\mu(x)}{\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\mu}/{d\mu}(x)}\xrightarrow{n\rightarrow +\infty} 0. \end{align*} $$

We say that $G\curvearrowright (X,\mu )$ is strongly conservative if there exists a sequence $(\eta _n)$ of probability measures on G that is strongly recurrent for $G\curvearrowright (X,\mu )$ .

Lemma 3.5. Let $G\curvearrowright (X,\mu )$ and $G\curvearrowright (Y,\nu )$ be non-singular actions of a countable infinite group G on standard probability spaces $(X,\mu )$ and $(Y,\nu )$ . Suppose that $\psi \colon (X,\mu )\rightarrow (Y,\nu )$ is a measure-preserving G-equivariant factor map and that $\eta _n$ is a sequence of probability measures on G that is strongly recurrent for the action ${G\curvearrowright (X,\mu )}$ . Then $\eta _n$ is strongly recurrent for the action $G\curvearrowright (Y,\nu )$ .

Proof. Let $E\colon L^0(X,[0,+\infty ))\rightarrow L^0(Y,[0,+\infty ))$ denote the conditional expectation map that is uniquely determined by

$$ \begin{align*} \int_Y E(F) H\, d\nu=\int_X F (H\circ \psi)\, d\mu \end{align*} $$

for all positive measurable functions $F\colon X\rightarrow [0,+\infty )$ and $H\colon Y\rightarrow [0,+\infty )$ . Since

$$ \begin{align*} \frac{dk^{-1}\nu}{d\nu}=\frac{d\psi_*(k^{-1}\mu)}{d\psi_*\mu}=E\bigg(\frac{dk^{-1}\mu}{d\mu}\bigg) \end{align*} $$

for every $k\in G$ , we have that

(3.6) $$ \begin{align} \sum_{k\in G}\eta_n(hk^{-1})\frac{dk^{-1}\nu}{d\nu}(y)=E\bigg(\sum_{k\in G}\eta_n(hk^{-1})\frac{dk^{-1}\mu}{d\mu}\bigg)(y) \quad\text{for a.e. } y\in Y. \end{align} $$

By Jensen’s inequality for conditional expectations, applied to the convex function ${t\mapsto 1/t}$ , we also have that

(3.7) $$ \begin{align} \frac{1}{E(\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\mu}/{d\mu})(y)}\kern1pt{\leq}\kern1pt E\kern-1pt\bigg(\frac{1}{\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\mu}/{d\mu}}\bigg)(y) \!\quad\text{for a.e. } y\kern1pt{\in}\kern1pt Y. \end{align} $$

Combining (3.6) and (3.7), we see that

$$ \begin{align*} &\sum_{h\in G}\eta_n^2(h)\int_{Y}\frac{d\nu(y)}{\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\nu}/{d\nu}(y)}\\ &\quad\leq \sum_{h\in G}\eta_n^2(h)\int_Y E\bigg(\frac{1}{\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\mu}/{d\mu}}\bigg)(y)\,d\nu(y)\\ &\quad=\sum_{h\in G}\eta_n^2(h)\int_X\frac{d\mu(x)}{\sum_{k\in G}\eta_n(hk^{-1}){dk^{-1}\mu}/{d\mu}(x)}, \end{align*} $$

which converges to $0$ as $\eta _n$ is strongly recurrent for $G\curvearrowright (X,\mu )$ .

We say that a non-singular group action $G\curvearrowright (X,\mu )$ has an invariant mean if there exists a G-invariant linear functional $\varphi \in L^{\infty }(X)^*$ . We say that $G\curvearrowright (X,\mu )$ is amenable (in the sense of Zimmer) if there exists a G-equivariant conditional expectation $E\colon L^{\infty }(G\times X)\rightarrow L^{\infty }(X)$ , where the action $G\curvearrowright G\times X$ is given by $g\cdot (h,x)=(gh,g\cdot x)$ .

Proposition 3.6. Let G be a countable infinite group and let $(\mu _g)_{g\in G }$ be a family of equivalent probability measures on a standard Borel space $X_0$ that is not supported on a single atom. Let $\nu $ be a probability measure on $X_0$ and for each $t\in [0,1]$ consider the Bernoulli action (3.2). Assume that $G\curvearrowright (X,\mu _1)$ is non-singular.

1. (1) If $G\curvearrowright (X,\mu _t)$ has an invariant mean, then $G\curvearrowright (X,\mu _s)$ has an invariant mean for every $s<t$ .

2. (2) If $G\curvearrowright (X,\mu _t)$ is amenable, then $G\curvearrowright (X,\mu _s)$ is amenable for every $s>t$ .

3. (3) If $G\curvearrowright (X,\mu _t)$ is strongly conservative, then $G\curvearrowright (X,\mu _s)$ is strongly conservative for every $s<t$ .

Proof. (1) We may assume that $t=1$ . So suppose that $G\curvearrowright (X,\mu _1)$ has an invariant mean and fix $s<1$ . Let $\unicode{x3bb} $ be the probability measure on $\{0,1\}$ that is given by ${\unicode{x3bb} (0)=s}$ . Then by [AIM19, Proposition A.9] the diagonal action $G\curvearrowright (X\times X\times \{0,1\}^{G}, \mu _1\times \mu _0\times \unicode{x3bb} ^{G})$ has an invariant mean. Since $G\curvearrowright (X,\mu _s)$ is a factor of this diagonal action, it admits a G-invariant mean as well.

(2) It suffices to show that $G\curvearrowright (X,\mu _1)$ is amenable whenever there exists a ${t\in (0,1)}$ such that $G\curvearrowright (X,\mu _t)$ is amenable. Write $\unicode{x3bb} $ for the probability measure on $\{0,1\}$ given by $\unicode{x3bb} (0)=t$ . Then $G\curvearrowright (X,\mu _t)$ is a factor of the diagonal action $G\curvearrowright (X\times X\times \{0,1\}^{G},\mu _1\times \mu _0\times \unicode{x3bb} ^{G})$ , so by [Zim78, Theorem 2.4] also the latter action is amenable. Since $G\curvearrowright (X\times \{0,1\}^{G},\mu _0\times \unicode{x3bb} ^{G})$ is pmp, we have that $G\curvearrowright (X,\mu _1)$ is amenable.

(3) We may again assume that $t=1$ . Suppose that $(\eta _n)$ is a strongly recurrent sequence of probability measures on G for the action $G\curvearrowright (X,\mu _1)$ . Fix $s<1$ and let $\unicode{x3bb} $ be the probability measure on $\{0,1\}$ defined by $\unicode{x3bb} (0)=s$ . As the diagonal action $G\curvearrowright (X\times \{0,1\}^{G},\mu _0\times \unicode{x3bb} ^{G})$ is pmp, the sequence $\eta _n$ is also strongly recurrent for the diagonal action $G\curvearrowright (X\times X\times \{0,1\},\mu _1\times \mu _0\times \unicode{x3bb} ^{G})$ . Since $G\curvearrowright (X,\mu _t)$ is a factor of $G\curvearrowright (X\times X\times \{0,1\}^{G},\mu _1\times \mu _0\times \unicode{x3bb} ^{G})$ , it follows from Lemma 3.5 that the sequence $\eta _n$ is strongly recurrent for $G\curvearrowright (X,\mu _t)$ .

We finally prove Theorem 3.3. The proof relies heavily upon the techniques developed in [MV20, §5].

Proof of Theorem 3.3

For every $t\in (0,1]$ write $\rho ^t$ for the Koopman representation

$$ \begin{align*} \rho^t\colon G\curvearrowright L^2(X,\mu_t):\;\;\;\; (\rho^t_g(\xi))(x)=\left(\frac{dg\mu_t}{d\mu_t}(x)\right)^{1/2}\xi(g^{-1}\cdot x). \end{align*} $$

Fix $s\in (0,1)$ and let $C>0$ be such that $\log (1-x)\geq -C x$ for every $x\in [0,s)$ . Then for every $t<s$ and every $g\in G$ we have that

$$ \begin{align*} \log(\langle \rho^t_g(1), 1\rangle)&=\sum_{h\in G}\log(1-H^2(\mu_{gh}^{t},\mu_h^{t}))\\ &\geq \sum_{h\in G}\log(1-tH^2(\mu_{gh},\mu_h))\\ &\geq -C t\sum_{h\in G}H^2(\mu_{gh},\mu_h). \end{align*} $$

Because $G\curvearrowright (X,\mu _1)$ is non-singular we get that

(3.8) $$ \begin{align} \langle \rho^t_g(1),1\rangle\rightarrow 1 \quad\text{as } t\rightarrow 0, \text{ for every } g\in G. \end{align} $$

We claim that there exists a $t'>0$ such that $G\curvearrowright (X,\mu _t)$ is non-amenable for every ${t<t'}$ . Suppose, to the contrary, that $t_n$ is a sequence that converges to zero such that ${G\curvearrowright (X,\mu _{t_n})}$ is amenable for every $n\in \mathbb {N}$ . Then it follows from [Nev03, Theorem 3.7] that $\rho ^{t_n}$ is weakly contained in the left regular representation $\unicode{x3bb} _G$ for every $n\in \mathbb {N}$ . Write $1_G$ for the trivial representation of G. It follows from (3.8) that $\bigoplus _{n\in \mathbb {N}}\rho ^{t_n}$ has almost invariant vectors, so that

$$ \begin{align*} 1_G\prec \bigoplus_{n\in \mathbb{N}}\rho^{t_n}\prec \infty \unicode{x3bb}_G\prec \unicode{x3bb}_G, \end{align*} $$

which is in contradiction to the non-amenability of G. By Theorem 3.1 there exists a ${t_1\in [0,1]}$ such that $G\curvearrowright (X,\mu _t)$ is weakly mixing for every $t<t_1$ . Since every dissipative action is amenable (see, for example, [AIM19, Theorem A.29]) it follows that $t_1\geq t'>0$ .

Write $Z_0=[0,1)$ and let $\unicode{x3bb} $ denote the Lebesgue probability measure on $Z_0$ . Let $\rho ^0$ denote the reduced Koopman representation

$$ \begin{align*} \rho^0\colon G\curvearrowright L^2(X\times Z_0^{G},\mu_0\times \unicode{x3bb}^G)\ominus \mathbb{C} 1: \quad (\rho^0_g(\xi))(x)=\xi(g^{-1}\cdot x). \end{align*} $$

As G is non-amenable, $\rho ^{0}$ has stable spectral gap. Suppose that for every $s>0$ we can find $0<s'<s$ such that $\rho ^{s'}$ is weakly contained in $\rho ^{s'}\otimes \rho ^{0}$ . Then there exists a sequence $s_n$ that converges to zero, such that $\rho ^{s_n}$ is weakly contained in $\rho ^{s_n}\otimes \rho ^{0}$ for every $n\in \mathbb {N}$ . This implies that $\bigoplus _{n\in \mathbb {N}}\rho ^{s_n}$ is weakly contained in $(\bigoplus _{n\in \mathbb {N}}\rho ^{s_n})\otimes \rho ^{0}$ . But by (3.8), the representation $\bigoplus _{n\in \mathbb {N}}\rho ^{s_n}$ has almost invariant vectors, so that $(\bigoplus _{n\in \mathbb {N}}\rho ^{s_n})\otimes \rho ^{0}$ weakly contains the trivial representation. This is in contradiction to $\rho ^{0}$ having stable spectral gap. We conclude that there exists an $s>0$ such that $\rho ^t$ is not weakly contained in $\rho ^t\otimes \rho ^0$ for every $t<s$ .

We prove that $G\curvearrowright (X,\mu _t)$ is strongly ergodic for every $t<\min \{t',s\}$ , in which case we can apply [MV20, Lemma 5.2] to the non-singular action $G\curvearrowright (X,\mu _t)$ and the pmp action $G\curvearrowright (X\times Z_0^{G},\mu _0\times \unicode{x3bb} ^G)$ by our choice of $t'$ and s. After rescaling, we may assume that $G\curvearrowright (X,\mu _1)$ is ergodic and that $\rho ^{t}$ is not weakly contained in $\rho ^{t}\otimes \rho ^{0}$ for every $t\in (0,1)$ .

Let $t\in (0,1)$ be arbitrary and define the map

$$ \begin{align*} \Psi\colon X\times X\times Z_0^{G}\rightarrow X:\quad \Psi(x,y,z)_h=\begin{cases}x_h &\text{if } z_h\leq t,\\ y_h &\text{if } z_h>t.\end{cases} \end{align*} $$

Then $\Psi $ is G-equivariant and we have that $\Psi (\mu _1\times \mu _0\times \unicode{x3bb} ^{G})=\mu _t$ . Suppose that ${G\curvearrowright (X,\mu _t)}$ is not strongly ergodic. Then we can find a bounded almost invariant sequence $f_n\in L^{\infty }(X,\mu _t)$ such that $\|f_n\|_2=1$ and $\mu _t(f_n)=0$ for every $n\in \mathbb {N}$ . Therefore, $\Psi _*(f_n)$ is a bounded almost invariant sequence for $G\curvearrowright (X\times X\times Z_0^{G},\mu _1\times \mu _0\times \unicode{x3bb} ^{G})$ . Let $E\colon L^{\infty }(X\times X\times Z_0^{G})\rightarrow L^{\infty }(X)$ be the conditional expectation that is uniquely determined by $\mu _1\circ E=\mu _1\times \mu _0\times \unicode{x3bb} ^{G}$ . By [MV20, Lemma 5.2] we have that $\lim _{n\rightarrow \infty }\|(E\circ \Psi _*)(f_n)-\Psi _*(f_n)\|_2=0$ . As $\Psi $ is measure-preserving we get, in particular, that

(3.9) $$ \begin{align} \lim_{n\rightarrow \infty}\|(E\circ \Psi_*)(f_n)\|_2=1. \end{align} $$

Note that if $\mu _t(f)=0$ for some $f\in L^{2}(X,\mu _t)$ , we have that $\mu _1((E\circ \Psi _*)(f))=0$ . So we can view $E\circ \Psi _*$ as a bounded operator

$$ \begin{align*} E\circ\Psi_*\colon L^2(X,\mu_t)\ominus \mathbb{C} 1\rightarrow L^2(X,\mu_1)\ominus \mathbb{C} 1. \end{align*} $$

Claim. The bounded operator $E\circ \Psi _*\colon L^2(X,\mu _t)\ominus \mathbb {C} 1\rightarrow L^2(X,\mu _1)\ominus \mathbb {C} 1$ has norm strictly less than $1$ .

The claim is in direct contradiction to (3.9), so we conclude that $G\curvearrowright (X,\mu _t)$ is strongly ergodic.

Proof of claim

For every $g\in G$ , let $\varphi _g$ be the map

$$ \begin{align*} \varphi_g\colon L^2(X_0,\mu_g^t)\rightarrow L^2(X_0,\mu_g): \quad \varphi_g(F)=tF+(1-t)\nu(F)\cdot 1. \end{align*} $$

Then $E\circ \Psi _*\colon L^2(X_0,\mu _t)\rightarrow L^2(X,\mu _1)$ is given by the infinite product $\bigotimes _{g\in G}\varphi _g$ . For every $g\in G$ we have that

$$ \begin{align*} \|F\|_{2,\mu_g}=\|(d\mu_g^t/d\mu_g)^{-1/2}F\|_{2,\mu_g^t}\leq t^{-1/2}\|F\|_{2,\mu_g^t}, \end{align*} $$

so that the inclusion map $\iota _g \colon L^2(X_0,\mu _g^t)\hookrightarrow L^2(X_0,\mu _g)$ satisfies $\|\iota _g\|\leq t^{-1/2}$ for every $g\in G$ . We have that

$$ \begin{align*} \varphi_g(F)=t(F-\mu_g(F)\cdot 1)+\mu_t(F)\cdot 1 \quad\text{for every } F\in L^2(X_0,\mu_g^t). \end{align*} $$

So if we write $P_g^t$ for the projection map onto $L^2(X_0,\mu _g^t)\ominus \mathbb {C} 1$ , and $P_g$ for the projection map onto $L^2(X_0,\mu _g)\ominus \mathbb {C} 1$ , we have that

(3.10) $$ \begin{align} \varphi_g\circ P^t_g=t(P_g\circ \iota_g)\quad\text{for every } g\in G. \end{align} $$

For a non-empty finite subset $\mathcal {F}\subset G$ let $V(\mathcal {F})$ be the linear subspace of $L^2(X,\mu _t)\ominus \mathbb {C} 1$ spanned by

$$ \begin{align*}\bigg(\bigotimes_{g\in \mathcal{F}}L^2(X_0,\mu_g^t)\ominus \mathbb{C} 1\bigg)\otimes \bigotimes_{g\in G\setminus \mathcal{F}}1. \end{align*} $$

Then, using (3.10), we see that

$$ \begin{align*} \|(E\circ \Psi_*)(f)\|_2\leq t^{|\mathcal{F}|/2}\|f\|_2 \quad\text{for every } f\in V(\mathcal{F}). \end{align*} $$

Since $\bigoplus _{\mathcal {F}\neq \emptyset }V(\mathcal {F})$ is dense inside $L^2(X,\mu _t)\ominus \mathbb {C} 1$ , we have that

$$ \begin{align*} \|(E\circ\Psi_*)|_{L^2(X,\mu_t)\ominus\mathbb{C} 1} \|\leq t^{1/2}<1.\\[-3.1pc] \end{align*} $$

This also concludes the proof of Theorem 3.3.

4 Non-singular Bernoulli actions arising from groups acting on trees: proof of Theorem C

Let T be a locally finite tree and choose a root $\rho \in T$ . Let $\mu _0$ and $\mu _1$ be equivalent probability measures on a standard Borel space $X_0$ . Following [AIM19, §10], we define a family of equivalent probability measures $(\mu _e)_{e\in E}$ by

(4.1) $$ \begin{align} \mu_e=\begin{cases}\mu_0 &\text{if } e \text{ is oriented towards } \rho,\\ \mu_1 &\text{if } e \text{ is oriented away from } \rho.\end{cases} \end{align} $$

Let $G\subset \operatorname {Aut}(T)$ be a subgroup. When $g\in G$ and $e\in E$ , the edges e and $g\cdot e$ are simultaneously oriented towards, or away from $\rho $ , unless $e\in E([\rho ,g\cdot \rho ])$ . As $E([\rho ,g\cdot \rho ])$ is finite for every $g\in G$ , the generalized Bernoulli action

(4.2) $$ \begin{align} G\curvearrowright (X,\mu)=\prod_{e\in E}(X_0,\mu_e): \quad (g\cdot x)_e=x_{g^{-1}\cdot e} \end{align} $$

is non-singular. If we start with a different root $\rho '\in T$ , let $(\mu ^{\prime }_e)_{e\in E}$ denote the corresponding family of probability measures on $X_0$ . Then we have that $\mu _e=\mu ^{\prime }_e$ for all but finitely many $e\in E$ , so that the measures $\prod _{e\in E}\mu _e$ and $\prod _{e\in E}\mu ^{\prime }_e$ are equivalent. Therefore, up to conjugacy, the action (4.2) is independent of the choice of root $\rho \in T$ .

Lemma 4.1. Let T be a locally finite tree such that each vertex $v\in V(T)$ has degree at least $2$ . Suppose that $G\subset \operatorname {Aut}(T)$ is a countable subgroup. Let $\mu _0$ and $\mu _1$ be equivalent probability measures on a standard Borel space $X_0$ and fix a root $\rho \in T$ . Then the action $\alpha \colon G\curvearrowright (X,\mu )$ given by (4.2) is essentially free.

Proof. Take $g\in G\setminus \{e\}$ . It suffices to show that $\mu (\{x\in X:g\cdot x=x\})=0$ . If g is elliptic, there exist disjoint infinite subtrees $T_1,T_2\subset T$ such that $g\cdot T_1=T_2$ . Note that

$$ \begin{align*} (X_1,\mu_1)=\prod_{e\in E(T_1)}(X_0,\mu_e) \quad\text{and}\quad (X_2,\mu_2)=\prod_{e\in E(T_2)}(X_0,\mu_e) \end{align*} $$

are non-atomic and that g induces a non-singular isomorphism $\varphi \colon (X_1,\mu _1)\rightarrow (X_2,\mu _2): \varphi (x)_e=x_{g^{-1}\cdot e}$ . We get that

$$ \begin{align*} \mu_1\times \mu_2(\{(x, \varphi(x)): x\in X_1\})=0. \end{align*} $$

A fortiori $\mu (\{x\in X:g\cdot x=x\})=0$ . If g is hyperbolic, let $L_g\subset T$ denote its axis on which it acts by non-trivial translation. Then $\prod _{e\in E(L_g)}(X_0,\mu _e)$ is non-atomic and by [BKV19, Lemma 2.2] the action $g^{\mathbb {Z}}\curvearrowright \prod _{e\in E(L_g)}(X_0,\mu _e)$ is essentially free. This implies that also $\mu (\{x\in X:g\cdot x=x\})=0$ .

We prove Theorem 4.2 below, which implies Theorem C and also describes the stable type when the action is weakly mixing.

Theorem 4.2. Let T be a locally finite tree with root $\rho \in T$ . Let $G\subset \operatorname {Aut}(T)$ be a closed non-elementary subgroup with Poincaré exponent $\delta =\delta (G\curvearrowright T)$ given by (1.5). Let $\mu _0$ and $\mu _1$ be non-trivial equivalent probability measures on a standard Borel space $X_0$ . Consider the generalized non-singular Bernoulli action $\alpha \colon G\curvearrowright (X,\mu )$ given by (4.2). Then $\alpha $ is:

• • weakly mixing if $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ ;

• • dissipative up to compact stabilizers if $1-H^2(\mu _0,\mu _1)<\exp (-\delta /2)$ .

Let $G\curvearrowright (Y,\nu )$ be an ergodic pmp action and let $\Lambda \subset \mathbb {R}$ be the smallest closed subgroup that contains the essential range of the map

$$ \begin{align*} X_0\times X_0\rightarrow \mathbb{R}: \quad (x,x')\mapsto \log(d\mu_0/d\mu_1)(x)-\log(d\mu_0/d\mu_1)(x'). \end{align*} $$

Let $\Delta \colon G\rightarrow \mathbb {R}_{>0}$ denote the modular function and let $\Sigma $ be the smallest subgroup generated by $\Lambda $ and $\log (\Delta (G))$ .

Suppose that $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ . Then the Krieger flow and the flow of weights of $\beta \colon G\curvearrowright X\times Y$ are determined by $\Lambda $ and $\Sigma $ as follows.

1. (1) If $\Lambda $ (respectively, $\Sigma $ ) is trivial, then the Krieger flow (respectively, flow of weights) is given by $\mathbb {R}\curvearrowright \mathbb {R}$ .

2. (2) If $\Lambda $ (respectively, $\Sigma $ ) is dense, then the Krieger flow (respectively, flow of weights) is trivial.

3. (3) If $\Lambda $ (respectively, $\Sigma $ ) equals $a\mathbb {Z}$ , with $a>0$ , then the Krieger flow (respectively, flow of weights) is given by $\mathbb {R}\curvearrowright \mathbb {R}/a\mathbb {Z}$ .

In general, we do not know the behaviour of the action (4.2) in the critical situation ${1-H^2(\mu _0,\mu _1)=\exp (-\delta /2)}$ . However, if T is a regular tree and $G\curvearrowright T$ has full Poincaré exponent, we prove in Proposition 4.3 below that the action is dissipative up to compact stabilizers. This is similar to [AIM19, Theorems 8.4 and 9.10].

Proposition 4.3. Let T be a q-regular tree with root $\rho \in T$ and let $G\subset \operatorname {Aut}(T)$ be a closed subgroup with Poincaré exponent $\delta =\delta (G\curvearrowright T)=\log (q-1)$ . Let $\mu _0$ and $\mu _1$ be equivalent probability measures on a standard Borel space $X_0$ .

If $1-H^2(\mu _0,\mu _1)=(q-1)^{-1/2}$ , then the action (4.2) is dissipative up to compact stabilizers.

Interesting examples of actions of the form (4.2) arise when $G\subset \operatorname {Aut}(T)$ is the free group on a finite set of generators acting on its Cayley tree. In that case, following [AIM19, §6] and [MV20, Remark 5.3], we can also give a sufficient criterion for strong ergodicity.

Proposition 4.4. Let the free group $\mathbb {F}_d$ on $d\geq 2$ generators act on its Cayley tree T. Let $\mu _0$ and $\mu _1$ be equivalent probability measures on a standard Borel space $X_0$ . Then the action (4.2) dissipative if $1-H^2(\mu _0,\mu _1)\leq (2d-1)^{-1/2}$ and weakly mixing and non-amenable if $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/2}$ . Furthermore, the action (4.2) is strongly ergodic when $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ .

The proof of Theorem 4.2 below is similar to that of [LP92, Theorem 4] and [AIM19, Theorems 10.3 and 10.4]

Proof of Theorem 4.2

Define a family $(X_e)_{e\in E}$ of independent random variables on $(X,\mu )=\prod _{e\in E}(X_0,\mu _e)$ by

(4.3) $$ \begin{align} X_e(x)=\begin{cases}\log (d\mu_1/d\mu_0) (x_e)&\text{if } e \text{ is oriented towards } \rho,\\ \log (d\mu_0/d\mu_1)(x_e) &\text{if } e \text{ is oriented away from } \rho.\end{cases} \end{align} $$

For $v\in T$ we write

$$ \begin{align*} S_v=\sum_{e\in E([\rho, v])} X_e. \end{align*} $$

Then we have that

$$ \begin{align*} \frac{dg\mu}{d\mu}=\exp(S_{g\cdot \rho})\quad\text{for every }g\in G. \end{align*} $$

Since $G\subset \operatorname {Aut}(T)$ is a closed subgroup, for each $v\in T$ the stabilizer subgroup $G_v=\{g\in G:g\cdot v= v\}$ is a compact open subgroup of G.

Suppose that $1-H^2(\mu _0,\mu _1)<\exp (-\delta /2)$ . Then we have that

$$ \begin{align*} \int_X\sum_{v\in G\cdot \rho}\exp(S_v(x)/2)\,d\mu(x)=\sum_{v\in G\cdot \rho}(1-H^2(\mu_0,\mu_1))^{2d(\rho,v)}<+\infty, \end{align*} $$

by definition of the Poincaré exponent. Therefore, we have that $\sum _{v\in G\cdot \rho }\exp (S_v(x)/2)<+\infty $ for a.e. $x\in X$ . Let $\unicode{x3bb} $ denote the left invariant Haar measure on G and define ${L=\unicode{x3bb} (G_\rho )}$ , where $G_\rho =\{g\in G:g\cdot \rho =\rho \}$ . Then we have that

$$ \begin{align*} \int_{G}\frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)=L\sum_{v\in G\cdot \rho }\exp(S_v(x))<+\infty\quad \text{for a.e. } x\in X. \end{align*} $$

We conclude that $G\curvearrowright (X,\mu )$ is dissipative up to compact stabilizers.

Now assume that $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ . We start by proving that $G\curvearrowright (X,\mu )$ is infinitely recurrent. By [AIM19, Theorem 8.17] we can find a non-elementary closed compactly generated subgroup $G'\subset G$ such that $1-H^2(\mu _0,\mu _1)>\exp (-\delta (G')/2)$ . Let $T'\subset T$ be the unique minimal $G'$ -invariant subtree. Then $G'$ acts cocompactly on $T'$ and we have that $\delta (G')=\dim _{H}\partial T'$ . Let X and Y be independent random variables with distributions $(\log d\mu _1/d\mu _0)_*\mu _0$ and $(\log d\mu _0/d\mu _1)_*\mu _1$ , respectively. Set $Z=X+Y$ and write

$$ \begin{align*} \varphi(t)=\mathbb{E}(\exp(tZ)). \end{align*} $$

The assignment $t\mapsto \varphi (t)$ is convex, $\varphi (t)=\varphi (1-t)$ for every t and $\varphi (1/2)= (1-H^2(\mu _0,\mu _1))^2$ . We conclude that

$$ \begin{align*} \inf_{t\geq 0}\varphi(t)=(1-H^2(\mu_0,\mu_1))^{2}. \end{align*} $$

Write $R_k$ for the sum of k independent copies of Z. By the Chernoff–Cramér theorem, as stated in [LP92], there exists an $M\in \mathbb {N}$ such that

(4.4) $$ \begin{align} \mathbb{P}(R_M\geq 0)>\exp(-M\delta(G')). \end{align} $$

Below we define a new unoriented tree S. This means that the edge set of S consists of subsets $\{v,w\}\subset V(S)$ . Fix a vertex $\rho '\in T'$ and define the unoriented tree S as follows.

• • S has vertices $v\in T'$ so that $d_{T'}(\rho ', v)$ is divisible by M.

• • There is an edge $\{v,w\}\in E(S)$ between two vertices $v,w\in S$ if $d_{T'}(v,w)=M$ and $[\rho ',v]_{T'}\subset [\rho ',w]_{T'}$ .

Here the notation $[\rho ',v]_{T'}$ means that we consider the line segment $[\rho ',v]$ as a subtree of $T'$ . We have that $\dim _H\partial S=M\dim _H \partial T'= M\delta (G')$ . Form a random subgraph $S(x)$ of S by deleting those edges $\{v,w\}\in E(S)$ where

$$ \begin{align*} \sum_{e\in E([v,w]_{T'})}X_e(x_e)<0. \end{align*} $$

This is an edge percolation on S, where each edge remains with probability ${p=\mathbb {P}(R_M\geq 0)}$ . So by (4.4) we have that $p\exp (\dim _H S)>1$ . Furthermore, if $\{v,w\}$ and $\{v',w'\}$ are edges of S so that $E([v,w]_{T'})\cap E([v',w']_{T'})=\emptyset $ , their presence in $S(x)$ constitutes independent events. So the percolation process is a quasi-Bernoulli percolation as introduced in [Lyo89]. Taking $w\in (1,p\exp (\dim _H S))$ and setting $w_n=w^{-n}$ , it follows from [Lyo89, Theorem 3.1] that percolation occurs almost surely, that is, $S(x)$ contains an infinite connected component for a.e. $x\in X$ . Writing

$$ \begin{align*} S^{\prime}_v(x)=\sum_{e\in E([\rho',v]_{T'})}X_e(x_e), \end{align*} $$

this means that for a.e. $x\in (X,\mu )$ we can find a constant $a_x>-\infty $ such that $S^{\prime }_v(x)>a_x$ for infinitely many $v\in T'$ . As $T'/G'$ is finite, there exists a vertex $w\in T'$ such that

(4.5) $$ \begin{align} \sum_{v\in G'\cdot w}\exp(S^{\prime}_v(x))=+\infty \quad\text{with positive probability.} \end{align} $$

Therefore, by Kolmogorov’s zero–one law, we have that $\sum _{v\in G'\cdot w}\exp (S^{\prime }_v(x))=+\infty $ almost surely. Since a change of root results in a conjugate action, we may assume that $\rho =w$ . Then (4.5) implies that $\sum _{v\in G\cdot \rho }\exp (S_v(x))=+\infty $ for a.e. $x\in X$ . Writing again L for the Haar measure of the stabilizer subgroup $G_\rho =\{g\in G:g\cdot \rho = \rho \}$ , we see that

$$ \begin{align*} \int_{G}\frac{dg\mu}{d\mu}\,d\unicode{x3bb}(g)=L\sum_{v\in G\cdot \rho}\exp(S_v)=+\infty\quad\text{almost surely.} \end{align*} $$

We conclude that $G\curvearrowright (X,\mu )$ is infinitely recurrent. We prove that $G\curvearrowright (X,\mu )$ is weakly mixing using a phase transition result from the previous section. Define the measurable map

$$ \begin{align*} \psi\colon X_0\rightarrow (0,1]:\quad \psi(x)=\min\{d\mu_1/d\mu_0(x),1\}. \end{align*} $$

Let $\nu $ be the probability measure on $X_0$ determined by

$$ \begin{align*} \frac{d\nu}{d\mu_0}(x)=\rho^{-1}\psi(x)\quad \text{where } \rho=\int_{X_0}\psi(x)\,d\mu_0(x). \end{align*} $$

Then we have that $\nu \sim \mu _0$ and for every $s>1-\rho $ the probability measures

$$ \begin{align*} \eta_0^s&=s^{-1}(\mu_0-(1-s)\nu),\\ \eta_1^s&=s^{-1}(\mu_1-(1-s)\nu) \end{align*} $$

are well defined. We consider the non-singular actions $G\curvearrowright (X,\eta _s)=\prod _{e\in E}(X_0,\eta _e^s)$ , where

$$ \begin{align*} \eta_e^s=\begin{cases}\eta_0^s &\text{if } e \text{ is oriented towards } \rho,\\ \eta_1^s &\text{if } e \text{ is oriented away from } \rho.\end{cases} \end{align*} $$

By the dominated convergence theorem we have that $H^2(\eta _0^s,\eta _1^s)\rightarrow H^2(\mu _0,\mu _1)$ as $s\rightarrow 1$ . So we can choose s close enough to $1$ , but not equal to $1$ , such that $1-H^2(\eta _0^s,\eta _1^s)>\exp (-\delta /2)$ . By the first part of the proof we have that $G\curvearrowright (X,\eta _s)$ is infinitely recurrent. Note that

$$ \begin{align*} \mu_j=(1-s)\nu+s\eta_j^s\quad\text{for } j=0,1. \end{align*} $$

Since we assumed that $G\subset \operatorname {Aut}(T)$ is closed, all the stabilizer subgroups $G_{v}=\{g\in G:g\cdot v=v\}$ are compact. By Remark 3.4 we conclude that $G\curvearrowright (X,\mu )$ is weakly mixing.

Let $G\curvearrowright (Y,\nu )$ be an ergodic pmp action. To determine the Krieger flow and the flow of weights of $\beta \colon G\curvearrowright X\times Y$ we use a similar approach to [AIM19, Theorem 10.4] and [VW17, Proposition 7.3]. First we determine the Krieger flow and then we deal with the flow of weights.

As before, let $G'\subset G$ be a non-elementary compactly generated subgroup such that ${1-H^2(\mu _0,\mu _1)>\exp (-\delta (G')/2)}$ . By [AIM19, Theorem 8.7] we may assume that $G/G'$ is not compact. Let $T'\subset T$ be the minimal $G'$ -invariant subtree. Let $v\in T'$ be as in Lemma 4.5 below so that

(4.6) $$ \begin{align} \bigcap_{g\in G} \left(E(gT')\cup E([v,g^{-1}\cdot v])\right)=\emptyset. \end{align} $$

Since changing the root yields a conjugate action, we may assume that $\rho =v$ . Let $(Z_0,\zeta _0)$ be a standard probability space such that there exist measurable maps $\theta _0,\theta _1\colon Z_0\rightarrow X_0$ that satisfy $(\theta _0)_*\zeta _0=\mu _0$ and $(\theta _1)_*\zeta _0=\mu _1$ . Write

$$ \begin{align*}(Z,\zeta)&=\prod_{e\in E(T)\setminus E(T')}(Z_0,\zeta_0),\\(X_1,\rho_1)&=\prod_{e\in E(T)\setminus E(T')}(X_0,\mu_e), \\(X_2,\rho_2)&=\prod_{e\in E(T')}(X_0,\mu_e).\end{align*} $$

By the first part of the proof we have that $G'\curvearrowright (X_2,\rho _2)$ is infinitely recurrent. Define the pmp map

$$ \begin{align*} \Psi\colon (Z,\zeta)\rightarrow (X_1,\rho_1):\;\;\;\; (\Psi(z))_e=\begin{cases}\theta_0(z_e) &\text{ if } e \text{ is oriented towards } \rho,\\ \theta_1(z_e) &\text{ if } e \text{ is oriented away from } \rho.\end{cases} \end{align*} $$

Consider

$$ \begin{align*} U=\{e\in E(T): e \text{ is oriented towards } \rho\}. \end{align*} $$

Since $gU\triangle U=E(T)([\rho ,g\cdot \rho ])\subset E(T')$ for any $g\in G'$ , the set $(E(T)\setminus E(T'))\cap U$ is $G'$ -invariant. Therefore, $\Psi $ is a $G'$ -equivariant factor map. Consider the Maharam extensions

$$ \begin{align*} G'\curvearrowright Z\times X_2\times Y\times \mathbb{R}\quad\text{and}\quad G\curvearrowright X\times Y\times \mathbb{R} \end{align*} $$

of the diagonal actions $G'\curvearrowright Z\times X_2\times Y$ and $G'\curvearrowright X\times Y\times \mathbb {R}$ , respectively. Identifying $(X,\mu )=(X_1,\rho _1)\times (X_2,\rho _2)$ , we obtain a $G'$ -equivariant factor map

$$ \begin{align*} \Phi\colon Z\times X_2\times Y\times \mathbb{R}\rightarrow X_1\times X_2\times Y\times \mathbb{R}:\quad \Phi(z,x,y,t)=(\Psi(z),x,y,t). \end{align*} $$

Take $F\in L^{\infty }(X\times Y\times \mathbb {R})^{G}$ . By [AIM19, Proposition A.33] the Maharam extension $G'\curvearrowright X_2\times Y\times \mathbb {R} $ is infinitely recurrent. Since $G'\curvearrowright Z$ is a mixing pmp generalized Bernoulli action we have that $F\circ \Phi \in L^{\infty }(Z\times X_2\times Y\times \mathbb {R})^{G}\subset 1\mathbin {\overline {\otimes }} L^{\infty }(X_2\times Y\times \mathbb {R})^{G}$ by [SW81, Theorem 2.3]. Therefore, F is essentially independent of the $E(T)\setminus E(T')$ -coordinates. Thus, for any $g\in G$ the assignment

$$ \begin{align*} (x,y,t)\mapsto F(g\cdot x,y,t)=F(x,y,t-\log(dg^{-1}\mu/d\mu)(x)) \end{align*} $$

is essentially independent of the $E(T)\setminus E(gT')$ -coordinates. Since $\log (dg^{-1}\mu /d\mu )$ only depends on the $E([\rho ,g^{-1}\cdot \rho ])$ -coordinates, we deduce that F is essentially independent of the $E(T)\setminus (E(gT')\cup E([\rho ,g^{-1}\cdot \rho ]))$ -coordinates, for every $g\in G$ . Therefore, by (4.6), we have that $F\in 1\mathbin {\overline {\otimes }} L^{\infty }(Y\times \mathbb {R})$ .

So we have proven that any G-invariant function $F\in L^{\infty }(X\times Y\times \mathbb {R})$ is of the form $F(x,y,t)=H(y,t)$ , for some $H\in L^{\infty }(Y\times \mathbb {R})$ that satisfies

$$ \begin{align*} H(y,t)=H(g\cdot y, t+\log(dg^{-1}\mu/d\mu)(x)) \quad\text{for a.e. } (x,y,t)\in X\times Y\times \mathbb{R}. \end{align*} $$

Since $0$ is in the essential range of the maps $\log (dg\mu /d\mu )$ , for every $g\in G$ , we see that $H(g\cdot y,t)=H(y,t)$ for a.e. $(y,t)\in Y\times \mathbb {R}$ . By ergodicity of $G\curvearrowright Y$ , we conclude that H is of the form $H(y,t)=P(t)$ , for some $P\in L^{\infty }(\mathbb {R})$ that satisfies

(4.7) $$ \begin{align} P(t)=P(t+\log(dg^{-1}\mu/d\mu)(x))\quad\text{for a.e. } (x,t)\in X\times \mathbb{R}, \text{ for every } g\in G. \end{align} $$

Let $\Gamma \subset \mathbb {R}$ be the subgroup generated by the essential ranges of the maps $\log (dg\mu /d\mu )$ , for $g\in G$ . If $\Gamma =\{0\}$ we can identify $L^{\infty }(X\times Y\times \mathbb {R})^{G}\cong L^{\infty }(\mathbb {R})$ . If $\Gamma \subset \mathbb {R}$ is dense, then it follows that P is essentially constant so that the Maharam extension $G\curvearrowright X\times Y\times \mathbb {R}$ is ergodic, that is, the Krieger flow of $G\curvearrowright X\times Y$ is trivial. If $\Gamma =a\mathbb {Z}$ , with $a>0$ , we conclude by (4.7) that we can identify $L^{\infty }(X\times Y\times \mathbb {R})^{G}\cong L^{\infty }(\mathbb {R}/a\mathbb {Z})$ , so that the Krieger flow of $G\curvearrowright X\times Y$ is given by $\mathbb {R}\curvearrowright \mathbb {R}/a\mathbb {Z}$ . Finally, note that the closure of $\Gamma $ equals the closure of the subgroup generated by the essential range of the map

$$ \begin{align*} X_0\times X_0\rightarrow \mathbb{R}\colon \quad (x,x')\mapsto \log(d\mu_0/d\mu_1)(x)-\log(d\mu_0/d\mu_1)(x'). \end{align*} $$

So we have calculated the Krieger flow in every case, concluding the proof of the theorem in the case where G is unimodular.

When G is not unimodular, let $G_0=\ker \Delta $ be the kernel of the modular function. Let $G\curvearrowright X\times Y\times \mathbb {R}$ be the modular Maharam extension and let $\alpha \colon G_0\curvearrowright X\times Y\times \mathbb {R}$ be its restriction to the subgroup $G_0$ . Then we have that

$$ \begin{align*} L^{\infty}(X\times Y\times \mathbb{R})^{G}\subset L^{\infty}(X\times Y \times \mathbb{R})^{\alpha}. \end{align*} $$

By [AIM19, Theorem 8.16] we have that $\delta (G_0)=\delta $ , and we can apply the argument above to conclude that $L^{\infty }(X\times Y\times \mathbb {R})^{\alpha }\subset 1\mathbin {\overline {\otimes }} 1\mathbin {\overline {\otimes }} L^{\infty }(\mathbb {R})$ . So for every $F\in L^{\infty }(X\times Y\times \mathbb {R})^{G}$ there exists a $P\in L^{\infty }(\mathbb {R})$ such that

(4.8) $$ \begin{align} P(t)\kern1.3pt{=}\kern1.3pt P(t\kern1.3pt{+}\log(dg^{-1}\mu/d\mu)(x)\kern1.3pt{+}\kern1.3pt\log(\Delta(g))) \!\quad\text{for a.e. } (x,t)\kern1.3pt{\in}\kern1.3pt X\kern1.3pt{\times}\kern1.3pt \mathbb{R}, \text{ for every } g\kern1.3pt{\in}\kern1.3pt G. \end{align} $$

Let $\Pi $ be the subgroup of $\mathbb {R}$ generated by the essential range of the maps

$$ \begin{align*} x\mapsto \log(dg^{-1}\mu/d\mu)(x)+\log(\Delta(g)) \quad\text{with } g\in G. \end{align*} $$

As $0$ is contained in the essential range of $\log (dg^{-1}\mu /d\mu )$ , for every $g\in G$ , we get that $\log (\Delta (G))\subset \Pi $ . Therefore, $\Pi $ also contains the subgroup $\Gamma \subset \mathbb {R}$ defined above. Thus, the closure of $\Pi $ equals the closure of $\Sigma $ , where $\Sigma \subset \mathbb {R}$ is the subgroup as in the statement of the theorem. From (4.8) we conclude that we may identify $L^{\infty }(X\times Y\times \mathbb {R})^{G}\cong L^{\infty }(\mathbb {R})^{\Sigma }$ , so that the flow of weights of $G\curvearrowright X\times Y$ is as stated in the theorem.

Lemma 4.5. Let T be a locally finite tree and let $G\subset \operatorname {Aut}(T)$ be a closed subgroup. Suppose that $H\subset G$ is a closed compactly generated subgroup that contains a hyperbolic element and assume that $G/H$ is not compact. Let $S\subset T$ be the unique minimal H-invariant subtree. Then there exists a vertex $v\in S$ such that

(4.9) $$ \begin{align} \bigcap_{g\in G}\left(gS\cup [v,g^{-1}\cdot v]\right)=\{v\}. \end{align} $$

Proof. Let $k\in H$ be a hyperbolic element and let $L\subset T$ be its axis, on which k acts by a non-trivial translation. Then $L\subset S$ , as one can show for instance as in the proof of [CM11, Proposition 3.8]. Pick any vertex $v\in L$ . We claim that this vertex will satisfy (4.9). Take any $w\in V(T)\setminus \{v\}$ . As $G/H$ is not compact, one can show as in [AIM19, Theorem 9.7] that there exists a $g\in G$ such that $g\cdot w\notin S$ . Since k acts by translation on L, there exists an $n\in \mathbb {N}$ large enough such that

$$ \begin{align*} [v,k\cdot v]\subset [v,k^ng\cdot v]\quad\text{and}\quad [v,k^{-1}\cdot v]\subset [v,k^{-n}g\cdot v], \end{align*} $$

so that in particular we have that $w\notin [v,k^ng\cdot v]\cap [v,k^{-n}g\cdot v]=\{v\}$ . Since S is H-invariant, we also have that $k^ng\cdot w\notin S$ and $k^{-n}g\cdot w\notin S$ and we conclude that

$$ \begin{align*} w\notin ((k^ng)^{-1}S\cup[v,k^ng\cdot v])\cap((k^{-n}g)^{-1}S\cup [v,k^{-n}g\cdot v]).\\[-3.1pc] \end{align*} $$

Proof of Proposition 4.3

Define the family $(X_e)_{e\in E}$ of independent random variables on $(X,\mu )$ by (4.3) and write

$$ \begin{align*} S_v=\sum_{e\in E([\rho,v])}X_e. \end{align*} $$

Claim. There exists a $\delta>0$ such that

$$ \begin{align*} \mu(\{x\in X:S_v(x)\leq -\delta\quad\text{for every } v\in T\setminus \{\rho\}\})>0. \end{align*} $$

Proof of claim

Note that $\mathbb {E}(\exp (X_e/2))=1-H^2(\mu _0,\mu _1)$ for every $e\in E$ . Define a family of random variables $(W_n)_{n\geq 0}$ on $(X,\mu )$ by

$$ \begin{align*} W_n=\sum_{\substack{v\in T\\d(v,\rho)=n}}\exp(S_v/2). \end{align*} $$

Using that $1-H^2(\mu _0,\mu _1)=(q-1)^{-1/2}$ , one computes that

$$ \begin{align*} \mathbb{E}(W_{n+1}|\;S_v,\; d(v,\rho)\leq n)=W_n\quad\text{for every } n\geq 1. \end{align*} $$

So the sequence $(W_n)_{n\geq 0}$ is a martingale, and since it is positive it converges almost surely to a finite limit when $n\rightarrow +\infty $ . Write $\Sigma _n=\{v\in T:d(v,\rho )=n\}$ . As ${W_n\geq \max _{v\in \Sigma _n}\exp (S_v/2)}$ we conclude that there exists a positive constant $C<+\infty $ such that

$$ \begin{align*} \mathbb{P}(S_v\leq C\text{ for every }v\in T)>0. \end{align*} $$

For any vertex $w\in T$ , write $T_w=\{v\in T:[\rho ,w]\subset [\rho ,v]\}$ : the set of children of w, including w itself. Using the symmetry of the tree and changing the root from $\rho $ to $w\in T$ , we also have that

(4.10) $$ \begin{align} \mathbb{P}(S_v-S_w\leq C\text{ for every } v\in T_w)>0\quad\text{for every } w\in T. \end{align} $$

Set $\nu _0=(\log d\mu _1/d\mu _0)_*\mu _0$ and $\nu _1=(\log d\mu _0/d\mu _1)_*\mu _1$ . Because $1-H^2(\mu _0,\mu _1)\neq 0$ we have that $\mu _0\neq \mu _1$ , so that there exists a $\delta>0$ such that

$$ \begin{align*} \nu_0*\nu_1((-\infty,-\delta))>0. \end{align*} $$

Here $\nu _0*\nu _1$ denotes the convolution product of $\nu _0$ with $\nu _1$ . Therefore, there exists $N\in \mathbb {N}$ large enough such that

(4.11) $$ \begin{align} \mathbb{P}(S_w\leq -C-\delta \text{ for every } w\in \Sigma_N \text{ and } S_{w'}\leq-\delta \text{ for every } w'\in \Sigma_n \text{ with } n\leq N)>0. \end{align} $$

Since for any $w\in \Sigma _N$ and $w'\in \Sigma _n$ with $n\leq N$ , we have that $S_v-S_w$ is independent of $S_{w'}$ for every $v\in T_w$ , and since $\Sigma _N$ is a finite set, it follows from (4.10) and (4.11) that

$$ \begin{align*} \mathbb{P}(S_v\leq -\delta \quad\text{for every } v\in T\setminus\{\rho\})>0. \end{align*} $$

This concludes the proof of the claim.

Let $\delta>0$ be as in the claim and define

$$ \begin{align*} \mathcal{U}=\{x\in X:S_v(x)\leq -\delta\text{ for every } v\in T\setminus \{\rho\}\}, \end{align*} $$

so that $\mu (\mathcal {U})>0$ . Let $G_\rho $ be the stabilizer subgroup of $\rho $ . Note that for every $g,h\in G$ we have that $S_{hg\cdot \rho }(x)=S_{g\cdot \rho }(h^{-1}\cdot x)+S_{h\cdot \rho }(x)$ for a.e. $x\in X$ , so that for $h\in G$ we have that

$$ \begin{align*} h\cdot \mathcal{U}\subset \{x\in X:S_{hg\cdot \rho}(x)\leq -\delta+S_{h\cdot \rho}(x)\text{ for every } g\notin G_{\rho}\}. \end{align*} $$

It follows that if $h\notin G_{\rho }$ , we have that

$$ \begin{align*} \mathcal{U}\cap h\cdot \mathcal{U}\subset \{x\in X:S_{h\cdot \rho}(x)\leq -\delta\text{ and }S_{h\cdot \rho}(x)\geq \delta\}=\emptyset. \end{align*} $$

Since $G\subset \operatorname {Aut}(T)$ is closed, we have that $G_\rho $ is compact. So the action $G\curvearrowright (X,\mu )$ is not infinitely recurrent. Let $\unicode{x3bb} $ denote the left invariant Haar measure on G. By an adaptation of the proof of [BV20, Proposition 4.3], the set

$$ \begin{align*} D=\bigg\{ x\in X: \int_{G}\frac{dg\mu}{d\mu}(x)\,d\unicode{x3bb}(g)<+\infty\bigg\}=\bigg\{ x\in X: \int_{G}\exp(S_{g\cdot \rho}(x))\,d\unicode{x3bb}(g)<+\infty\bigg\} \end{align*} $$

satisfies $\mu (D)\in \{0,1\}$ . Since $G\curvearrowright (X,\mu )$ is not infinitely recurrent, it follows from [AIM19, Proposition A.28] that $\mu (D)>0$ , so that we must have that $\mu (D)=1$ . By [AIM19, Theorem A.29] the action $G\curvearrowright (X,\mu )$ is dissipative up to compact stabilizers.

We use a similar approach to [MV20, §6] in the proof of Proposition 4.4.

Proof of Proposition 4.4

It follows from Theorem 4.2 and Proposition 4.3 that the action $G\curvearrowright (X,\mu )$ , given by (4.2), is dissipative when $1-H^2(\mu _0,\mu _1)\leq (2d-1)^{-1/2}$ and weakly mixing when $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/2}$ . So it remains to show that $G\curvearrowright (X,\mu )$ is non-amenable when $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/2}$ and strongly ergodic when $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ .

Assume first that $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/2}$ . By taking the kernel of a surjective homomorphism $\mathbb {F}_d\rightarrow \mathbb {Z}$ we find a normal subgroup $H_1\subset \mathbb {F}_d$ that is free on infinitely many generators. By [RT13, Théorème 0.1] we have that $\delta (H_1)=(2d-1)^{-1/2}$ . Then, using [Sul79, Corollary 6], we can find a finitely generated free subgroup $H_2\subset H_1$ such that $H_1=H_2*H_3$ for some free subgroup $H_3\subset H_1$ and such that $1-H^2(\mu _0,\mu _1)>\exp (-\delta (H_2)/2)$ . Let $\psi \colon H_1\rightarrow H_3$ be the surjective group homomorphism uniquely determined by

$$ \begin{align*} \psi(h)=\begin{cases}e &\text{if } h\in H_2,\\ h&\text{if } h\in H_3.\end{cases} \end{align*} $$

We set $N=\ker \psi $ , so that $H_2\subset N$ and we get that $1-H^2(\mu _0,\mu _1)>\exp (-\delta (N)/2)$ . Therefore, $N\curvearrowright (X,\mu )$ is ergodic by Theorem 4.2. Also we have that $H_1/N\cong H_3$ , which is a free group on infinitely many generators. Therefore, $H_1\curvearrowright (X,\mu )$ is non-amenable by [MV20, Lemma 6.4]. A posteriori also $\mathbb {F}_d\curvearrowright (X,\mu )$ is non-amenable.

Let $\pi $ be the Koopman representation of the action $\mathbb {F}_d\curvearrowright (X,\mu )$ :

$$ \begin{align*} \pi\colon G\curvearrowright L^2(X,\mu):\quad (\pi_g(\xi))(x)=\left(\frac{dg\mu}{d\mu}(x)\right)^{1/2}\xi(g^{-1}\cdot x). \end{align*} $$

Claim. If $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ , then $\pi $ is not weakly contained in the left regular representation.

Proof of claim

Let $\eta $ denote the canonical symmetric measure on the generator set of $\mathbb {F}_d$ and define

$$ \begin{align*} P=\sum_{g\in \mathbb{F}_d}\eta(g)\pi_g. \end{align*} $$

The $\eta $ -spectral radius of $\alpha \colon \mathbb {F}_d\curvearrowright (X,\mu )$ , which we denote by $\rho _\eta (\alpha )$ , is by definition the norm of P, as a bounded operator on $L^2(X,\mu )$ . By [AIM19, Proposition A.11] we have that

$$ \begin{align*} \rho_\eta(\alpha)&=\lim_{n\rightarrow \infty}\langle P^n(1), 1\rangle^{1/n}\\ &=\lim_{n\rightarrow \infty}\bigg(\sum_{g\in \mathbb{F}_d}\eta^{*n}(g)(1-H^2(\mu_0,\mu_1))^{2|g|}\bigg)^{1/n}, \end{align*} $$

where $|g|$ denotes the word length of a group element $g\in \mathbb {F}_d$ . By [AIM19, Theorem 6.10] we then have that

$$ \begin{align*} \rho_\eta(\alpha)=\frac{(1-H^2(\mu_0,\mu_1))^2}{2d}\left((2d-1)+(1-H^2(\mu_0,\mu_1))^{-4}\right) \end{align*} $$

if $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ , and

$$ \begin{align*} \rho_\eta(\alpha)= \frac{\sqrt{2d-1}}{d} \end{align*} $$

if $1-H^2(\mu _0,\mu _1)\leq (2d-1)^{-1/4}$ . Therefore, if $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ , we have that $\rho _\eta (\alpha )>\rho _\eta (\mathbb {F}_d)$ , where $\rho _\eta (\mathbb {F}_d)$ denotes the $\eta $ -spectral radius of the left regular representation. This implies that $\alpha $ is not weakly contained in the left regular representation (see, for instance, [AD03, §3.2]).

Now assume that $1-H^2(\mu _0,\mu _1)>(2d-1)^{-1/4}$ . As in the proof of Theorem 4.2 there exist probability measures $\nu , \eta _0$ and $\eta _1$ on $X_0$ that are equivalent to $\mu _0$ and a number $s\in (0,1)$ such that

$$ \begin{align*} \mu_j=(1-s)\nu+s\eta_j\quad\text{for } j=0,1, \end{align*} $$

and such that $1-H^2(\eta _0,\eta _1)>(2d-1)^{-1/4}$ . Consider the non-singular action

$$ \begin{align*} \mathbb{F}_d\curvearrowright (X,\eta)=\prod_{e\in E(T)}(X_0,\eta_e)\quad \text{where } \eta_e=\begin{cases}\eta_0&\text{if } e \text{ is oriented towards } \rho,\\ \eta_1 &\text{if } e \text{ is oriented away from } \rho.\end{cases} \end{align*} $$

By Theorem 4.2 the action $\mathbb {F}_d\curvearrowright (X,\eta )$ is ergodic. Write $\rho $ for the Koopman representation associated to $\mathbb {F}_d\curvearrowright (X,\eta )$ . By the claim, $\rho $ is not weakly contained in the left regular representation. Let $\unicode{x3bb} $ be the probability measure on $\{0,1\}$ given by ${\unicode{x3bb} (0)=s}$ . Let $\rho ^0$ be the reduced Koopman representation of the pmp generalized Bernoulli action ${\mathbb {F}_d\curvearrowright (X\times \{0,1\}^{E(T)},\nu ^{E(T)}\times \unicode{x3bb} ^{E(T)})}$ . Then $\rho ^0$ is contained in a multiple of the left regular representation. Therefore, as $\rho $ is not weakly contained in the left regular representation, $\rho $ is not weakly contained in $\rho \otimes \rho ^{0}$ .

Define the map

$$ \begin{align*} \Psi\colon X\times X\times \{0,1\}^{E(T)}\rightarrow X\colon \;\;\;\; \Psi(x,y,z)_e=\begin{cases}x_e &\text{if } z_e=0,\\ y_e &\text{if } z_e=1.\end{cases} \end{align*} $$

Then $\Psi $ is $\mathbb {F}_d$ -equivariant and we have that $\Psi _*(\eta \times \nu ^{E(T)}\times \unicode{x3bb} ^{E(T)})=\mu $ . Suppose that $\mathbb {F}_d\curvearrowright (X,\mu )$ is not strongly ergodic. Then there exists a bounded almost invariant sequence $f_n\in L^{\infty }(X,\mu )$ such that $\|f_n\|_2=1$ and $\mu (f_n)=0$ for every $n\in \mathbb {N}$ . Therefore, $\Psi _*(f_n)$ is a bounded almost invariant sequence for the diagonal action ${\mathbb {F}_d\curvearrowright (X\times X\times \{0,1\}^{E(T)},\eta \times \nu ^{E(T)}\times \unicode{x3bb} ^{E(T)})}$ . Let $E\colon L^{\infty }(X\times X\times \{0,1\}^{E(T)})\rightarrow L^{\infty }(X)$ be the conditional expectation that is uniquely determined by $\mu \circ E=\eta \times \nu ^{E(T)}\times \unicode{x3bb} ^{E(T)}$ . By [MV20, Lemma 5.2] we have that $\lim _{n\rightarrow \infty }\|(E\circ \Psi _*)(f_n)-\Psi _{*}(f_n)\|_2=0$ , and in particular we get that

(4.12) $$ \begin{align} \lim_{n\rightarrow \infty}\|(E\circ \Psi_*)(f_n)\|_2=1. \end{align} $$

But just as in the proof of Theorem 3.3 we have that

$$ \begin{align*} \left\|(E\circ\Psi_*)\big|_{L^2(X,\mu)\ominus \mathbb{C}1}\right\|<1, \end{align*} $$

which is in contradiction with (4.12). We conclude that $\mathbb {F}_d\curvearrowright (X,\mu )$ is strongly ergodic.

Proposition 4.6 below complements Theorem 4.2 by considering groups $G\subset \operatorname {Aut}(T)$ that are not closed. This is similar to [AIM19, Theorem 10.5].

Proposition 4.6. Let T be a locally finite tree with root $\rho \in T$ . Let $G\subset \operatorname {Aut}(T)$ be an lcsc group such that the inclusion map $G\rightarrow \operatorname {Aut}(T)$ is continuous and such that ${G\subset \operatorname {Aut}(T)}$ is not closed. Write $\delta =\delta (G\curvearrowright T)$ for the Poincaré exponent given by (1.5). Let $\mu _0$ and $\mu _1$ be non-trivial equivalent probability measures on a standard Borel space $X_0$ . Consider the generalized non-singular Bernoulli action $\alpha \colon G\curvearrowright (X,\mu )$ given by (4.2). Let $H\subset \operatorname {Aut}(T)$ be the closure of G. Then the following assertions hold.

• • If $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ , then $\alpha $ is ergodic and its Krieger flow is determined by the essential range of the map

  (4.13) $$ \begin{align} X_0\times X_0\rightarrow \mathbb{R}:\quad (x,x')\mapsto\log(d\mu_0/d\mu_1)(x)-\log(d\mu_0/d\mu_1)(x') \end{align} $$
  as in Theorem 4.2.

• • If $1-H^2(\mu _0,\mu _1)<\exp (-\delta /2)$ , then each ergodic component of $\alpha $ is of the form $G\curvearrowright H/K$ , where K is a compact subgroup of H. In particular, there exists a G-invariant $\sigma $ -finite measure on X that is equivalent to $\mu $ .

Proof. Let $H\subset \operatorname {Aut}(T)$ be the closure of G. Then $\delta (H)=\delta $ and we can apply Theorem 4.2 to the non-singular action $H\curvearrowright (X,\mu )$ .

If $1-H^2(\mu _0,\mu _1)>\exp (-\delta /2)$ , then $H\curvearrowright X$ is ergodic. As $G\subset H$ is dense, we have that

$$ \begin{align*} L^{\infty}(X)^{G}=L^{\infty}(X)^{H}=\mathbb{C}1, \end{align*} $$

so that $G\curvearrowright X$ is ergodic. Let $H\curvearrowright X\times \mathbb {R}$ be the Maharam extension associated to  ${H\curvearrowright X}$ . Again, as $G\subset H$ is dense, we have that

$$ \begin{align*} L^{\infty}(X\times \mathbb{R})^{G}=L^{\infty}(X\times \mathbb{R})^{H}. \end{align*} $$

Note that the subgroup generated by the essential ranges of the maps $\log (dg^{-1}\mu /d\mu )$ , with $g\in G$ , is the same as the subgroup generated by the essential ranges of the maps $\log (dh^{-1}\mu /d\mu )$ , with $h\in H$ . Then one determines the Krieger flow of $G\curvearrowright X$ as in the proof of Theorem 4.2.

If $1-H^2(\mu _0,\mu _1)<\exp (-\delta /2)$ , the action $H\curvearrowright (X,\mu )$ is dissipative up to compact stabilizers. By [AIM19, Theorem A.29] each ergodic component is of the form $H\curvearrowright H/K$ for a compact subgroup $K\subset H$ . Therefore, each ergodic component of $G\curvearrowright (X,\mu )$ is of the form $G\curvearrowright H/K$ , for some compact subgroup $K\subset H$ .