Phase transitions for non-singular Bernoulli actions

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 [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference Meesschaert, Raum and VaesMRV11, 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 [Reference Schmidt and WaltersSW81, 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 [Reference Schmidt and WaltersSW81, Theorem 2.3] demands proper ergodicity of the action $G\curvearrowright (X,\mu )$ , the infinite recurrence assumption is sufficient as remarked in [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference SauvageotSa74, 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 [Reference SauvageotSa74, 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 ${G\curvearrowright X}$ .

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 [Reference KakutaniKak48] 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 [Reference Berendschot and VaesBKV19, 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 [Reference TitsTit70, 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 [Reference Caprace and de MedtsCM11, §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 $\delta (G\curvearrowright T)=\dim _{H}\partial T$ .