1 Introduction
When G is a countable infinite group and $(X_0,\mu _0)$ is a nontrivial standard probability space, the probability measurepreserving (pmp) action
is called a Bernoulli action. Probability measurepreserving Bernoulli actions are among the beststudied objects in ergodic theory and they play an important role in operator algebras [Reference IoanaIoa10, Reference PopaPop03, Reference PopaPop06]. 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
is in general no longer measurepreserving. Instead, we are interested in the case where $G\curvearrowright (X,\mu )$ is nonsingular, 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 nonsingular if and only if $\mu _h\sim \mu _g$ for every $h,g\in G$ and
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 nonsingular Bernoulli actions, determining conservativeness and ergodicity is much more difficult (see, for instance, [Reference Berendschot and VaesBKV19, Reference DanilenkoDan18, Reference KosloffKos18, Reference Vaes and WahlVW17]).
Besides nonsingular Bernoulli actions, another interesting class of nonsingular group actions comes from the Gaussian construction, as introduced in [Reference Arano, Isono and MarrakchiAIM19]. 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 1cocycle for the representation $\pi $ , then the assignment
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 nonsingular 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 [Reference Björklund, Kosloff and VaesBV20, 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 nonsingular 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 1cocycle $c\colon G\rightarrow \mathcal {H}$ with a parameter $t\in [0,+\infty )$ we get a oneparameter family of nonsingular 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 [Reference Arano, Isono and MarrakchiAIM19], we study a similar phase transition framework, but in the setting of nonsingular Bernoulli actions. Such a phase transition framework for nonsingular Bernoulli actions was already considered by Kosloff and Soo in [Reference Kosloff and SooKS20]. They showed the following phase transition result for the family of nonsingular Bernoulli actions of $G=\mathbb {Z}$ with base space $X_0=\{0,1\}$ that was introduced in [Reference Vaes and WahlVW17, Corollary 6.3]. For every $t\in [0,+\infty )$ consider the family of measures $(\mu _n^t)_{n\in \mathbb {Z}}$ given by
Then $\mathbb {Z}\curvearrowright (X,\mu _t)=\prod _{n\in \mathbb {Z}}(\{0,1\},\mu _n^t)$ is nonsingular 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 $t>t_1$ [Reference Kosloff and SooKS20, Theorem 3]. In [Reference Danilenko, Kosloff and RoyDKR20, Example D] the authors describe a family of nonsingular Poisson suspensions for which a similar phase transition occurs. These examples arise from dissipative essentially free actions of $\mathbb {Z}$ , and thus they are nonsingular Bernoulli actions. We generalize the phase transition result from [Reference Kosloff and SooKS20] to arbitrary nonsingular 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
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 nonsingular. 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
is nonsingular 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 nonamenable 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 nonsingular Bernoulli action is strongly ergodic. This is inspired by [Reference Arano, Isono and MarrakchiAIM19, Theorem 7.20] and [Reference Marrakchi and VaesMV20, Theorem 5.1], which state similar results for nonsingular Gaussian actions.
Theorem B. Let G be a countable infinite nonamenable group and suppose that the Bernoulli action $G\curvearrowright (X,\mu _1)=\prod _{g\in G}(X_0,\mu _g)$ is nonsingular. 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 [Reference Arano, Isono and MarrakchiAIM19, §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 [Reference Arano, Isono and MarrakchiAIM19, §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 $1p$ . If $G\subset \operatorname {Aut}(T)$ is a subgroup, then we naturally obtain a nonsingular 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
where $v\in V(T)$ is any vertex of T. In [Reference Arano, Isono and MarrakchiAIM19, Theorem 10.4] Arano, Isono and Marrakchi showed that if $G\subset \operatorname {Aut}(T)$ is a closed nonelementary subgroup, the action $G\curvearrowright (\Omega (T),\mu _p)$ is dissipative up to compact stabilizers if $2\sqrt {p(1p)}<\exp (\delta )$ and weakly mixing if $2\sqrt {p(1p)}>\exp (\delta )$ . 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
Suppose that $G\subset \operatorname {Aut}(T)$ is a subgroup. Then the generalized Bernoulli action
is nonsingular and up to conjugacy it does not depend on the choice of root $\rho \in T$ . In our next main result we generalize [Reference Arano, Isono and MarrakchiAIM19, Theorem 10.4] to nonsingular 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 nonelementary 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 $1H^2(\mu _0,\mu _1)<\exp (\delta /2)$ and weakly mixing if $1H^2(\mu _0,\mu _1)>\exp (\delta /2)$ .
2 Preliminaries
2.1 Nonsingular group actions
Let $(X,\mu ), (Y,\nu )$ be standard measure spaces. A Borel map $\varphi \colon X\rightarrow Y$ is called nonsingular 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 nonsingular isomorphism. We write $\operatorname {Aut}(X,\mu )$ for the group of all nonsingular 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 nonsingular 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 nonsingular 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 Ginvariant Borel set $A\subset X$ satisfies $\mu (A)=0$ or $\mu (X\setminus A)=0$ . A nonsingular 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
Suppose that $G\curvearrowright (X,\mu )$ is a nonsingular action and that $\mu $ is a probability measure. A sequence of Borel subsets $A_n\subset X$ is called almost invariant if
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 nonsingular 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 nonsingular 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 nonsingular action ${G\curvearrowright (X,\mu )}$ , with $\mu (X)=1$ , is dissipative up to compact stabilizers if and only if
where $\unicode{x3bb} $ denotes the left invariant Haar measure on G. We say that $G\curvearrowright (X,\mu )$ is infinitely recurrent if for every nonnegligible 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 nonsingular action $G\curvearrowright (X,\mu )$ , with $\mu (X)=1$ , is infinitely recurrent if and only if
A nonsingular 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 nonsingular 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 nonsingular action that is infinitely recurrent. Then we have that
Proof. Note that the set
is Ginvariant. 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
is a Borel subset of Y and we have that $\nu (\bigcup _{n\geq 1 } U_n)=1$ . Therefore, the sets
are analytic and exhaust X up to a set of measure zero. So there exist an $n_0\in \mathbb {N}$ and a nonnegligible 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 nonsingular 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
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 nonsingular automorphism $\varphi \in \operatorname {Aut}(X,\mu )$ , we define its Maharam extension by
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 nonsingular group action $G\curvearrowright (X,\mu )$ , by composing with this map, we obtain a nonsingular group action $G\curvearrowright X\times \mathbb {R}$ , which we call the Maharam extension of $G\curvearrowright X$ . If $G\curvearrowright X$ is a nonsingular 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 welldefined 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 Ginvariant $\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
In particular, if $G\curvearrowright (X,\mu )$ is a nonsingular 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
for $H\in L^{\infty }(X)$ and $h\in G$ . If $G\curvearrowright X$ is nonsingular 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 nonunimodular 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 nonsingular action. We define the modular Maharam extension of $G\curvearrowright X$ as the nonsingular action
Let $L^{\infty }(X\times \mathbb {R})^{\beta }$ denote the subalgebra of $\beta $ invariant elements. We define the flow of weights associated to $G{\kern1pt}\curvearrowright{\kern1pt} X$ as the translation action $\mathbb {R}{\kern1pt}\curvearrowright{\kern1pt} L^{\infty }(X{\kern1pt}\times{\kern1pt} \mathbb {R})^{\beta }: (t\cdot H)(x,s)= H(x,st)$ .
As we explain below, the flow of weights associated to an essentially free ergodic nonsingular 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 nonsingular 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
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
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
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
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 nonsingular 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 nonsingular 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 nonatomic and finite, we say that $G\curvearrowright X$ is of type II $_{1}$ . If $\nu $ is nonatomic 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 Nonsingular 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
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
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 nonsingular if and only if
If $(X,\mu )$ is nonatomic and the Bernoulli action (2.1) is nonsingular, 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
is called the generalized Bernoulli action and it is nonsingular 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
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 biinfinite line segment $L\subset T$ , called the axis of g, such that g acts on L by nontrivial 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 nonelementary 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 nonelementary 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 nonelementary closed subgroup, there exists a unique minimal Ginvariant 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 $\delta (G\curvearrowright T)=\dim _{H}\partial T$ .
3 Phase transitions of nonsingular 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
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 nonsingular. Let $\nu $ also be a probability measure on $X_0$ . Then for every $t\in [0,1]$ the Bernoulli action
is nonsingular. Assume, in addition, that one of the following conditions holds.

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

(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 nonsingular Bernoulli action. Note that Theorem 3.1 implies that
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 nonsingular with nonnegligible 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 $G\curvearrowright (X,\mu _t)= \prod _{g\in G}(X_0,(1t)\eta _e+t\mu _g)$ is constructed in the same way, by starting with the conservative Bernoulli action $G\curvearrowright \prod _{g\in G}(C_0,(1t)\eta _e+t\eta _g)$ . So for every $t\in (0,1)$ the Bernoulli action $G\curvearrowright (X,\mu _t)$ has nonnegligible conservative part and nonnegligible 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 nonamenable group. Make the same assumptions as in Theorem 3.1 and consider the nonsingular Bernoulli actions ${G\curvearrowright (X,\mu _t)}$ given by (3.2). Assume, moreover, that:

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

(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 nonsingular. For every $t\in [0,1]$ we have that
so that $G\curvearrowright (X,\mu _t)$ is nonsingular 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
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
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 Gequivariant factor map
Then the map $\mathord {\textrm {id}}_Y\times \Psi \colon Y\times X\times X\times Z\rightarrow Y\times X$ is Gequivariant 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 [Reference Kosloff and SooKS20, §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 [Reference Schmidt and WaltersSW81, Theorem 2.3] we have that
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 nonnegligible 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 Gequivariant 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 nonsingular 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
We shall write $\gamma _1=\gamma _1^1, \gamma _2=\gamma _2^1$ . Also define
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
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
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
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
As $\pi _*((\gamma _1\times \gamma _2^s) _{\mathcal {U}'})\sim \gamma _2^s _{W}$ , we have that
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
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
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
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
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
It follows from [Reference Björklund, Kosloff and VaesBV20, 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
is mixing. For $t\in [0,1]$ write
and assume that the generalized Bernoulli action $G\curvearrowright (X,\mu _1)$ is nonsingular.
Since [Reference Schmidt and WaltersSW81, Theorem 2.3] still applies to infinitely recurrent actions of lcsc groups (see [Reference Arano, Isono and MarrakchiAIM19, 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 [Reference Berendschot and VaesBKV19, Definition 4.2]. When G is a countable infinite group and $G\curvearrowright (X,\mu )$ is a nonsingular 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
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 nonsingular 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 measurepreserving Gequivariant 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
for all positive measurable functions $F\colon X\rightarrow [0,+\infty )$ and $H\colon Y\rightarrow [0,+\infty )$ . Since
for every $k\in G$ , we have that
By Jensen’s inequality for conditional expectations, applied to the convex function ${t\mapsto 1/t}$ , we also have that
Combining (3.6) and (3.7), we see that
which converges to $0$ as $\eta _n$ is strongly recurrent for $G\curvearrowright (X,\mu )$ .
We say that a nonsingular group action $G\curvearrowright (X,\mu )$ has an invariant mean if there exists a Ginvariant 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 Gequivariant 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 nonsingular.

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

(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 [Reference Arano, Isono and MarrakchiAIM19, 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 Ginvariant 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 [Reference ZimmerZim78, 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 [Reference Marrakchi and VaesMV20, §5].
Proof of Theorem 3.3.
For every $t\in (0,1]$ write $\rho ^t$ for the Koopman representation
Fix $s\in (0,1)$ and let $C>0$ be such that $\log (1x)\geq C x$ for every $x\in [0,s)$ . Then for every $t<s$ and every $g\in G$ we have that
Because $G\curvearrowright (X,\mu _1)$ is nonsingular we get that
We claim that there exists a $t'>0$ such that $G\curvearrowright (X,\mu _t)$ is nonamenable 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 [Reference NevoNev03, 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
which is in contradiction to the nonamenability 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, [Reference Arano, Isono and MarrakchiAIM19, 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
As G is nonamenable, $\rho ^{0}$ 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 [Reference Marrakchi and VaesMV20, Lemma 5.2] to the nonsingular 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
Then $\Psi $ is Gequivariant 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 [Reference Marrakchi and VaesMV20, Lemma 5.2] we have that $\lim _{n\rightarrow \infty }\(E\circ \Psi _*)(f_n)\Psi _*(f_n)\_2=0$ . As $\Psi $ is measurepreserving we get, in particular, that
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
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
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
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
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
For a nonempty 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
Then, using (3.10), we see that
Since $\bigoplus _{\mathcal {F}\neq \emptyset }V(\mathcal {F})$ is dense inside $L^2(X,\mu _t)\ominus \mathbb {C} 1$ , we have that
This also concludes the proof of Theorem 3.3.
4 Nonsingular 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 [Reference Arano, Isono and MarrakchiAIM19, §10], we define a family of equivalent probability measures $(\mu _e)_{e\in E}$ by
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
is nonsingular. 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
are nonatomic and that g induces a nonsingular isomorphism $\varphi \colon (X_1,\mu _1)\rightarrow (X_2,\mu _2): \varphi (x)_e=x_{g^{1}\cdot e}$ . We get that
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 nontrivial translation. Then $\prod _{e\in E(L_g)}(X_0,\mu _e)$ is nonatomic and by [Reference Berendschot and VaesBKV19, 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 nonelementary subgroup with Poincaré exponent $\delta =\delta (G\curvearrowright T)$ given by (1.5). Let $\mu _0$ and $\mu _1$ be nontrivial equivalent probability measures on a standard Borel space $X_0$ . Consider the generalized nonsingular Bernoulli action $\alpha \colon G\curvearrowright (X,\mu )$ given by (4.2). Then $\alpha $ is:

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

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