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.


Introduction
When G is a countable infinite group and (X 0 , μ 0 ) is a non-trivial standard probability space, the probability measure-preserving (pmp) action 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 (μ g ) g∈G on the base space X 0 that need not all be equal, the Bernoulli action is in general no longer measure-preserving.Instead, we are interested in the case where G (X, μ) is non-singular, that is, the group G preserves the measure class of μ.By Kakutani's criterion for equivalence of infinite product measures the Bernoulli action (1.1) is non-singular if and only if μ h ∼ μ g for every h, g ∈ G and 2

T. Berendschot
It is well known that a pmp Bernoulli action G (X 0 , μ 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 π : G → O(H) is an orthogonal representation of a locally compact second countable (lcsc) group on a real Hilbert space H, and if c : G → H is a 1-cocycle for the representation π , then the assignment α g (ξ ) = π g (ξ ) + c(g) (1.3) defines an affine isometric action α : G H. To any affine isometric action α : G H Arano, Isono and Marrakchi associated a non-singular group action α : G H, where H is the Gaussian probability space associated to H. When α : G 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 π : G → 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 g∈G (R, ν F (g) ), where F : G → R is such that c g (h) = F (g −1 h) − F (h), and ν F (g) denotes the Gaussian probability measure with mean F (g) and variance 1.
By scaling the 1-cocycle c : G → H with a parameter t ∈ [0, +∞) we get a one-parameter family of non-singular actions α t : G H t associated to the affine isometric actions α t : G H, given by α t g (ξ ) = π g (ξ ) + tc(g).Arano, Isono and Marrakchi showed that there exists a t diss ∈ [0, +∞) such that α t is dissipative up to compact stabilizers for every t > t diss and infinitely recurrent for every t < t 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 = Z with base space X 0 = {0, 1} that was introduced in [VW17, Corollary 6.3].For every t ∈ [0, +∞) consider the family of measures (μ t n ) n∈Z given by Then Z (X, μ t ) = n∈Z ({0, 1}, μ t n ) is non-singular for every t ∈ [0, +∞).Kosloff and Soo showed that there exists a t 1 ∈ (1/6, +∞) such that Z (X, μ 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 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.
https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press Phase transitions for non-singular Bernoulli actions 3 Suppose that G is a countable infinite group and let (μ g ) g∈G be a family of equivalent probability measure on a standard Borel space X 0 .Let ν also be a probability measure on X 0 .For every t ∈ [0, 1] we consider the family of equivalent probability measures (μ t g ) g∈G that are defined by μ t g = (1 − t)ν + tμ g .
( 1 .4 ) 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 (X, μ 1 ) = g∈G (X 0 , μ g ) is non-singular.Let ν ∼ μ e be a probability measure on X 0 and for every t ∈ [0, 1] consider the family (μ t g ) g∈G of equivalent probability measures given by (1.4).Then the Bernoulli action is non-singular for every t ∈ [0, 1] and there exists a t 1 ∈ [0, 1] such that G (X, μ 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 , μ 0 ), the pmp Bernoulli action G (X 0 , μ 0 ) G is strongly ergodic.Consider again the family of probability measures (μ t g ) g∈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 (X, μ 1 ) = g∈G (X 0 , μ g ) is non-singular.Let ν ∼ μ e be a probability measure on X 0 and for every t ∈ [0, 1] consider the family (μ t g ) g∈G of equivalent probability measures given by (1.4).Then there exists a t 0 ∈ (0, 1] such that G (X, μ t ) = g∈G (X 0 , μ t g ) 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 ⊂ 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 (T ) denote the set of orientations on T. Let p ∈ (0, 1) and fix a root ρ ∈ T .Define a probability measure μ p on (T ) by orienting an edge towards ρ with probability p and away from ρ with probability 1 − p.If G ⊂ Aut(T ) is a subgroup, then we naturally obtain a non-singular action G ( (T ), μ p ). Up to equivalence of measures, the measure μ p does not depend on the choice of root ρ ∈ T .The Poincaré exponent of G ⊂ Aut(T ) is defined as Let E(T ) ⊂ V (T ) × V (T ) denote the set of oriented edges, so that vertices v and w are adjacent if and only if (v, w), (w, v) ∈ E(T ).Suppose that X 0 is a standard Borel space and that μ 0 , μ 1 are equivalent probability measures on X 0 .Fix a root ρ ∈ T and define a family of probability measures (μ e ) e∈E(T ) by is non-singular and up to conjugacy it does not depend on the choice of root ρ ∈ T .In our next main result we generalize [AIM19, Theorem 10.4] to non-singular actions of the form (1.7).

Preliminaries
2.1.Non-singular group actions.Let (X, μ), (Y , ν) be standard measure spaces.A Borel map ϕ : X → Y is called non-singular if the pushforward measure ϕ * μ is equivalent to ν.If in addition there exist conull Borel sets X 0 ⊂ X and Y 0 ⊂ Y such that ϕ : X 0 → Y 0 is a bijection we say that ϕ is a non-singular isomorphism.We write Aut(X, μ) for the group of all non-singular automorphisms ϕ : X → X, where we identify two elements if they agree almost everywhere.The group Aut(X, μ) carries a canonical Polish topology.
A non-singular group action G (X, μ) of an lcsc group G on a standard measure space (X, μ) is a continuous group homomorphism G → Aut(X, μ).A non-singular group action G (X, μ) is called essentially free if the stabilizer subgroup G x = {g ∈ G : g • x = x} is trivial for almost every (a.e.) x ∈ X.When G is countable this is the same as the condition that μ({x ∈ X : g  Proof.Note that the set is G-invariant.Therefore, it suffices to show that G X is not infinitely recurrent under the assumption that D has full measure.

T. Berendschot
Let π : (X, μ) → (Y , ν) be the projection onto the space of ergodic components of G X. Then there exist a conull Borel subset Y 0 ⊂ Y and a Borel map θ : Y 0 → X such that (π • θ )(y) = y for every y ∈ Y 0 .
Write X y = π −1 ({y}).By [AIM19, Theorem A.29], for a.e.y ∈ Y there exists a compact subgroup be an increasing sequence of compact subsets of G such that n≥1 • G n = G.For every x ∈ X, write G x = {g ∈ G : g • x = x} for the stabilizer subgroup of x.Using an argument as in [MRV11, Lemma 10], one shows that for each n ≥ 1 the set {x ∈ X : G x ⊂ G n } is Borel.Thus, for every n ≥ 1 the set is a Borel subset of Y and we have that ν( n≥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 ∈ N and a non-negligible Borel set Then there exist y ∈ U n 0 and g 1 , We will frequently use the following result of Schmidt and Walters.Suppose that G (X, μ) is a non-singular action that is infinitely recurrent and suppose that G (Y , ν) is pmp and mixing.Then by [SW81, Theorem 2.3] we have that where G X × Y acts diagonally.Although [SW81, Theorem 2.3] demands proper ergodicity of the action G (X, μ), the infinite recurrence assumption is sufficient as remarked in [AIM19, Remark 7.4].

The Maharam extension and crossed products.
Let (X, μ) be a standard measure space.For any non-singular automorphism ϕ ∈ Aut(X, μ), we define its Maharam extension by Then ϕ preserves the infinite measure μ × exp(−t)dt.The assignment ϕ → ϕ is a continuous group homomorphism from Aut(X) to Aut(X × R).Thus, for each non-singular group action G (X, μ), by composing with this map, we obtain a non-singular group action G X × R, which we call the Maharam extension of G X. If G X is a non-singular group action, the translation action R X × R in the second component commutes with the Maharam extension G X × R. Therefore, we get a well-defined action R L ∞ (X × R) G , which is the Krieger flow associated to the action G X. The Krieger flow is given by R R if and only if there exists a G-invariant σ -finite measure ν on X that is equivalent to μ.
is the von Neumann algebra generated by the operators X is non-singular essentially free and ergodic, then L ∞ (X) G is a factor.Moreover, when G is a unimodular group, the Krieger flow of G X equals the flow of weights of the crossed product von Neumann algebra L ∞ (X) 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 : G → R >0 .Let λ denote the Lebesgue measure on R. Suppose that α : G (X, μ) is a non-singular action.We define the modular Maharam extension of G X as the non-singular action As we explain below, the flow of weights associated to an essentially free ergodic non-singular action G X equals the flow of weights of the crossed product factor L ∞ (X) G, justifying the terminology.See also [Sa74,Proposition 4.1].
Let α : G X be an essentially free ergodic non-singular group action with modular Maharam extension β : G X × R. By [Sa74, Proposition 1.1] there is a canonical normal semifinite faithful weight ϕ on L ∞ (X) α G such that the modular automorphism group σ ϕ is given by for the map given by ξ g,x (s) = ξ(s, g, x).Then by Fubini's theorem ξ g,x ∈ L 2 (R) for a.e.
Then one can verify that conjugates the dual action σ ϕ : R (L ∞ (X) α G) σ ϕ R and γ .Therefore, we can identify the flow of weights R Remark 2.3.It will be useful to speak about the Krieger type of a non-singular ergodic action G 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 ∞ (X) G.So assume that G is countable and that G (X, μ) is a non-singular ergodic action.Then the Krieger flow is ergodic and we distinguish several cases.If ν is atomic, we say that G X is of type I.If ν is non-atomic and finite, we say that G X is of type II 1 .If ν is non-atomic and infinite, we say that G X is of type II ∞ .If the Krieger flow is given by R R/ log(λ)Z with λ ∈ (0, 1), we say that G X is of type III λ .If the Krieger flow is the trivial flow R { * }, we say that G X is of type III 1 .If the Krieger flow is properly ergodic (that is, every orbit has measure zero), we say that G X is of type III 0 .

Non-singular Bernoulli actions.
Suppose that G is a countable infinite group and that (μ g ) g∈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 ν, η on a standard Borel space Y, the Hellinger distance H 2 (ν, η) is defined by Suppose that I is a countable infinite set and that (μ i ) i∈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 When ν is a probability measure on X 0 such that μ i = ν for every i ∈ I , the generalized Bernoulli action (2.3) is pmp and it is mixing if and only if the stabilizer subgroup 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 ) × V (T ) with the property that vertices v, w ∈ V (T ) are adjacent if and only if (v, w), (w, v) ∈ 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 ∈ T let [v, w] denote the smallest subtree of T that contains v and w.The distance between vertices v, w ∈ T is defined as Fixing a root ρ ∈ T , we define the boundary ∂T of T as the collection of all infinite line segments starting at ρ.We equip ∂T with a metric d ρ as follows.If ω, ω ∈ ∂T , let v ∈ T be the unique vertex such that d(ρ, v) = sup v∈ω∩ω d(ρ, v) and define Then, up to homeomorphism, the space (∂T , d ρ ) does not depend on the chosen root ρ ∈ T .Furthermore, the Hausdorff dimension dim H ∂T of (∂T , d ρ ) is also independent of the choice of ρ ∈ T .Let Aut(T ) denote the group of automorphisms of T. By [Tit70, Proposition 3.2], if g ∈ 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 ⊂ 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 Aut(T ) with the topology of pointwise convergence.A subgroup G ⊂ Aut(T ) is closed with respect to this topology if and only if for every v ∈ T the stabilizer subgroup

Phase transitions of non-singular Bernoulli actions: proof of Theorems A and B
Let G be a countable infinite group and let (μ g ) g∈G be a family of equivalent probability measures on a standard Borel space X 0 .Let ν also be a probability measure on X 0 .For t ∈ [0, 1] we define the family of probability measures T. Berendschot We write μ t for the infinite product measure μ t = g∈G μ t g on X = g∈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 (μ g ) g∈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 g∈G (X 0 , μ g ) is non-singular.Let ν also be a probability measure on X 0 .Then for every t ∈ [0, 1] the Bernoulli action is non-singular.Assume, in addition, that one of the following conditions holds.
(2) ν ≺ μ e and sup g∈G |log dμ g /dμ e (x)| < +∞ for a.e x ∈ X 0 .Then there exists a t 1 ∈ [0, 1] such that G (X, μ 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 ν ≺ μ e , and not the additional assumption that sup g∈G |log dμ g /dμ e (x)| < +∞ for a.e.x ∈ X 0 .However, the following example shows that this is not possible.
Let G be any countable infinite group and let G g∈G (C 0 , η g ) be a conservative non-singular Bernoulli action.Note that Theorem 3.1 implies that G g∈G (C 0 , (1 − t)η e + tη g ) is conservative for every t < 1.Let C 1 be a standard Borel space and let (μ g ) g∈G be a family of equivalent probability measures on X 0 = C 0 C 1 such that 0 < g∈G μ g (C 1 ) < +∞ and such that Proof of Claim 1.Note that for every g ∈ G we have that so that (μ s ) r = μ sr .Therefore, it suffices to prove that G (X, μ s ) is weakly mixing for every s < 1, assuming that G (X, μ 1 ) is conservative.The claim is trivially true for s = 0.So assume that G (X, μ 1 ) is conservative and fix s ∈ (0, 1).Let G (Y , η) be an ergodic pmp action.Define Y 0 = X 0 × X 0 × {0, 1} and define the probability measures λ on {0, 1} by λ(0) = s.Define the map θ : Y 0 → X 0 by θ(x, x , j) = x if j = 0, Then for every g ∈ G we have that θ * (μ g × ν × λ) = μ s g .Write Z = {0, 1} G and equip Z with the probability measure λ G .We identify the Bernoulli action G Y G 0 with the diagonal action G X × X × Z.By applying θ in each coordinate we obtain a G-equivariant factor map ) can be identified with a pmp Bernoulli action with base space (X 0 × {0, 1}, ν × λ), so that it is mixing.By [SW81, Theorem 2.3] we have that which implies that the assignment (y, x, x , z) → F (y, (x, x , z)) is essentially independent of x and z.Choosing a finite set of coordinates F ⊂ G and changing, for g ∈ F, the value z g between 0 and 1, we see that F is essentially independent of the x g -coordinates for g ∈ F. As this is true for any finite set F ⊂ G, we have that Proof of Claim 2. Again it suffices to assume that G (X, μ 1 ) is not dissipative and to show that G (X, μ s ) is conservative for every s < 1.When s = 0, the statement is trivial, so assume that G (X, μ 1 ) is not dissipative and fix s ∈ (0, 1).Let C ⊂ X denote the non-negligible conservative part of G (X, μ 1 ).As in the proof of Claim 1, write Z = {0, 1} G and let λ be the probability measure on {0, 1} given by λ(0) = s.Writing : X × X × Z → X for the G-equivariant map (3.4).We claim that * ((μ 1 × μ 0 × λ G )| C×X×Z ) ∼ μ s , so that G (X, μ s ) is a factor of a conservative non-singular action, and therefore must be conservative itself. As Let U ⊂ X be the Borel set, uniquely determined up to a set of measure zero, such that * ((μ We shall write By applying the map (3.3) in every coordinate, we get factor maps j : Y j → X j that satisfy Let π : X 1 × X 2 → X 2 and π : Y 1 × X 2 → X 2 denote the coordinate projections.Note that by construction we have that For every y ∈ X 2 define the Borel sets U y = {x ∈ X 1 : (x, y) ∈ U } and U y = {x ∈ X 1 : (x, y) ∈ U }.The disintegration of (γ As γ 1 (U y ) > 0 for γ s 2 -a.e.y ∈ W , and using that ν ∼ μ e , we see that Since this is true for every finite subset F ⊂ G, we conclude that μ s (X \ U ) = 0.
and assume that the generalized Bernoulli action G (X, μ 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 (X, μ t ) is infinitely recurrent, then G (X, μ 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 ∈ G : g • i = i} is compact for every i ∈ I and that ν ∼ μ e .Then there exists a t 1 ∈ [0, 1] such that G (X, μ 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 (X, μ) is a non-singular action on a standard probability space, a sequence (η n ) of probability measures on G is called strongly recurrent for the action We say that G (X, μ) is strongly conservative if there exists a sequence (η n ) of probability measures on G that is strongly recurrent for G (X, μ).
LEMMA 3.5.Let G (X, μ) and G (Y , ν) be non-singular actions of a countable infinite group G on standard probability spaces (X, μ) and (Y , ν).Suppose that ψ : (X, μ) → (Y , ν) is a measure-preserving G-equivariant factor map and that η n is a sequence of probability measures on G that is strongly recurrent for the action G (X, μ).Then η n is strongly recurrent for the action G (Y , ν).
We say that a non-singular group action G (X, μ) has an invariant mean if there exists a G-invariant linear functional ϕ ∈ L ∞ (X) * .We say that G (X, μ) is amenable (in the sense of Zimmer) if there exists a G-equivariant conditional expectation , where the action G G × X is given by g • (h, x) = (gh, g • x).PROPOSITION 3.6.Let G be a countable infinite group and let (μ g ) g∈G be a family of equivalent probability measures on a standard Borel space X 0 that is not supported on a single atom.Let ν be a probability measure on X 0 and for each t ∈ [0, 1] consider the Bernoulli action (3.2).Assume that G (X, μ 1 ) is non-singular.
(1) If G (X, μ t ) has an invariant mean, then G (X, μ s ) has an invariant mean for every s < t. , μ s ) is strongly conservative for every s < t.
Proof.(1) We may assume that t = 1.So suppose that G (X, μ 1 ) has an invariant mean and fix s < 1.Let λ be the probability measure on {0, 1} that is given by λ(0) = s.Then by [AIM19, Proposition A.9] the diagonal action G (X × X × {0, 1} G , μ 1 × μ 0 × λ G ) has an invariant mean.Since G (X, μ s ) is a factor of this diagonal action, it admits a G-invariant mean as well.
Proof of Theorem 3.3.For every t ∈ (0, 1] write ρ t for the Koopman representation Fix s ∈ (0, 1) and let C > 0 be such that log(1 − x) ≥ −Cx for every x ∈ [0, s).Then for every t < s and every g ∈ G we have that Because G (X, μ 1 ) is non-singular we get that ρ t g (1), 1 → 1 as t → 0, for every g ∈ G. (3.8) We claim that there exists a t > 0 such that G (X, μ 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 (X, μ t n ) is amenable for every n ∈ N. Then it follows from [Nev03, Theorem 3.7] that ρ t n is weakly contained in the left regular representation λ G for every n ∈ N. Write 1 G for the trivial representation of G.It follows from (3.8) that n∈N ρ t n has almost invariant vectors, so that which is in contradiction to the non-amenability of G.By Theorem 3.1 there exists a t 1 ∈ [0, 1] such that G (X, μ 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 ≥ t > 0.
https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press Phase transitions for non-singular Bernoulli actions 17 Write Z 0 = [0, 1) and let λ denote the Lebesgue probability measure on Z 0 .Let ρ 0 denote the reduced Koopman representation As G is non-amenable, ρ 0 has stable spectral gap.Suppose that for every s > 0 we can find 0 < s < s such that ρ s is weakly contained in ρ s ⊗ ρ 0 .Then there exists a sequence s n that converges to zero, such that ρ s n is weakly contained in ρ s n ⊗ ρ 0 for every n ∈ N.This implies that n∈N ρ s n is weakly contained in ( n∈N ρ s n ) ⊗ ρ 0 .But by (3.8), the representation n∈N ρ s n has almost invariant vectors, so that ( n∈N ρ s n ) ⊗ ρ 0 weakly contains the trivial representation.This is in contradiction to ρ 0 having stable spectral gap.We conclude that there exists an s > 0 such that ρ t is not weakly contained in ρ t ⊗ ρ 0 for every t < s.
We prove that G (X, μ 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 (X, μ t ) and the pmp action G (X × Z G 0 , μ 0 × λ G ) by our choice of t and s.After rescaling, we may assume that G (X, μ 1 ) is ergodic and that ρ t is not weakly contained in ρ t ⊗ ρ 0 for every t ∈ (0, 1).
Let t ∈ (0, 1) be arbitrary and define the map Then is G-equivariant and we have that is not strongly ergodic.Then we can find a bounded almost invariant sequence f n ∈ L ∞ (X, μ t ) such that f n 2 = 1 and μ t (f n ) = 0 for every n ∈ N.
The claim is in direct contradiction to (3.9), so we conclude that G (X, μ t ) is strongly ergodic.

Proof of claim.
For every g ∈ G, let ϕ g be the map https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press

T. Berendschot
Then E • * : L 2 (X 0 , μ t ) → L 2 (X, μ 1 ) is given by the infinite product g∈G ϕ g .For every g ∈ G we have that F 2,μ g = (dμ t g /dμ g ) −1/2 F 2,μ t g ≤ t −1/2 F 2,μ t g , so that the inclusion map ι g : L 2 (X 0 , μ t g ) → L 2 (X 0 , μ g ) satisfies ι g ≤ t −1/2 for every g ∈ G.We have that So if we write P t g for the projection map onto L 2 (X 0 , μ t g ) C1, and P g for the projection map onto L 2 (X 0 , μ g ) C1, we have that ϕ g • P t g = t (P g • ι g ) for every g ∈ G.
Then, using (3.10), we see that Since F =∅ V (F) is dense inside L 2 (X, μ t ) C1, we have that This also concludes the proof of Theorem 3.3.

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 ρ ∈ T .Let μ 0 and μ 1 be equivalent probability measures on a standard Borel space X 0 .Following [ are non-atomic and that g induces a non-singular isomorphism ϕ : (X 1 , μ 1 ) → (X 2 , μ 2 ) : ϕ(x) e = x g −1 •e .We get that If g is hyperbolic, let L g ⊂ T denote its axis on which it acts by non-trivial translation.Then e∈E(L g ) (X 0 , μ e ) is non-atomic and by [BKV19, Lemma 2.2] the action g Z e∈E(L g ) (X 0 , μ e ) is essentially free.This implies that also μ({x ∈ X : g 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 ρ ∈ T .Let G ⊂ Aut(T ) be a closed non-elementary subgroup with Poincaré exponent δ = δ(G T ) given by (1.5).Let μ 0 and μ 1 be non-trivial equivalent probability measures on a standard Borel space X 0 .Consider the generalized non-singular Bernoulli action α : G (X, μ) given by (4.2).Then α is: Let G (Y , ν) be an ergodic pmp action and let ⊂ R be the smallest closed subgroup that contains the essential range of the map Let : G → R >0 denote the modular function and let be the smallest subgroup generated by and log( (G)).
Suppose that 1 − H 2 (μ 0 , μ 1 ) > exp(−δ/2).Then the Krieger flow and the flow of weights of β : G X × Y are determined by and as follows.In general, we do not know the behaviour of the action (4.2) in the critical situation 1 − H 2 (μ 0 , μ 1 ) = exp(−δ/2).However, if T is a regular tree and G 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 ρ ∈ T and let G ⊂ Aut(T ) be a closed subgroup with Poincaré exponent δ = δ(G T ) = log(q − 1).Let μ 0 and μ 1 be equivalent probability measures on a standard Borel space X 0 .
Interesting examples of actions of the form (4.2) arise when G ⊂ Aut(T ) is the free group on a finite set of generators acting on its Cayley tree.In that case, following [ For v ∈ T we write Then we have that dgμ dμ = exp(S g•ρ ) for every g ∈ G. Since Suppose that 1 − H 2 (μ 0 , μ 1 ) < exp(−δ/2).Then we have that Since 0 is in the essential range of the maps log(dgμ/dμ), for every g ∈ G, we see that H (g • y, t) = H (y, t) for a.e.(y, t) ∈ Y × R. By ergodicity of G Y , we conclude that H is of the form H (y, t) = P (t), for some P ∈ L ∞ (R) that satisfies Let ⊂ R be the subgroup generated by the essential ranges of the maps log(dgμ/dμ), for g ∈ G.
Finally, note that the closure of equals the closure of the subgroup generated by the essential range of the map 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 be the kernel of the modular function.Let G X × Y × R be the modular Maharam extension and let α : G 0 X × Y × R be its restriction to the subgroup G 0 .Then we have that By [AIM19, Theorem 8.16] we have that δ(G 0 ) = δ, and we can apply the argument above to conclude that L (4.8) Let be the subgroup of R generated by the essential range of the maps x → log(dg −1 μ/dμ)(x) + log( (g)) with g ∈ G.
As 0 is contained in the essential range of log(dg −1 μ/dμ), for every g ∈ G, we get that log( (G)) ⊂ .Therefore, also contains the subgroup ⊂ R defined above.Thus, the closure of equals the closure of , where ⊂ R is the subgroup as in the statement of the theorem.From (4.8) we conclude that we may identify Proof.Let k ∈ H be a hyperbolic element and let L ⊂ T be its axis, on which k acts by a non-trivial translation.Then L ⊂ S, as one can show for instance as in the proof of [CM11, Proposition 3.8].Pick any vertex v ∈ L. We claim that this vertex will satisfy (4.9).Take any w ∈ V (T ) \ {v}.As G/H is not compact, one can show as in [AIM19, Theorem 9.7] that there exists a g ∈ G such that g • w / ∈ S. Since k acts by translation on L, there exists an n ∈ N large enough such that so that in particular we have that w / Since S is H-invariant, we also have that k n g • w / ∈ S and k −n g • w / ∈ S and we conclude that Proof of Proposition 4.3.Define the family (X e ) e∈E of independent random variables on (X, μ) by (4.3) and write CLAIM.There exists a δ > 0 such that μ({x ∈ X : S v (x) ≤ −δ for every v ∈ T \ {ρ}}) > 0.
Proof of claim.Note that E(exp(X e /2)) = 1 − H 2 (μ 0 , μ 1 ) for every e ∈ E. Define a family of random variables (W n ) n≥0 on (X, μ) by Using that 1 − H 2 (μ 0 , μ 1 ) = (q − 1) −1/2 , one computes that So the sequence (W n ) n≥0 is a martingale, and since it is positive it converges almost surely to a finite limit when n → +∞.Write n = {v ∈ T : d(v, ρ) = n}.As W n ≥ max v∈ n exp(S v /2) we conclude that there exists a positive constant C < +∞ such that P(S v ≤ C for every v ∈ T ) > 0.
By Theorem 4.2 the action F d (X, η) is ergodic.Write ρ for the Koopman representation associated to F d (X, η).By the claim, ρ is not weakly contained in the left regular representation.Let λ be the probability measure on {0, 1} given by λ(0) = s.Let ρ 0 be the reduced Koopman representation of the pmp generalized Bernoulli action F d (X × {0, 1} E(T ) , ν E(T ) × λ E(T ) ).Then ρ 0 is contained in a multiple of the left regular representation.Therefore, as ρ is not weakly contained in the left regular representation, ρ is not weakly contained in ρ ⊗ ρ 0 .
Define the map : X × X × {0, 1} E(T ) → X : (x, y, z) e = x e if z e = 0, Then is F d -equivariant and we have that * (η × ν E(T ) × λ E(T ) ) = μ.Suppose that F d (X, μ) is not strongly ergodic.Then there exists a bounded almost invariant sequence f n ∈ L ∞ (X, μ) such that f n 2 = 1 and μ(f n ) = 0 for every n ∈ N. Therefore, * (f n ) is a bounded almost invariant sequence for the diagonal action F d (X × X × {0, 1} E(T ) , η × ν E(T ) × λ E(T ) ).Let E : L ∞ (X × X × {0, 1} E(T ) ) → L ∞ (X) be the conditional expectation that is uniquely determined by μ • E = η × ν E(T ) × λ E (T ) .By [MV20, Lemma 5.2] we have that lim n→∞ (E • * )(f n ) − * (f n ) 2 = 0, and in particular we get that lim n→∞ (E • * )(f n ) 2 = 1. (4.12) But just as in the proof of Theorem 3.3 we have that which is in contradiction with (4.12).We conclude that F d (X, μ) is strongly ergodic.Proposition 4.6 below complements Theorem 4.2 by considering groups G ⊂ 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 ρ ∈ T .Let G ⊂ Aut(T ) be an lcsc group such that the inclusion map G → Aut(T ) is continuous and such that

( 1 )
If (respectively, ) is trivial, then the Krieger flow (respectively, flow of weights) is given by R R. (2) If (respectively, ) is dense, then the Krieger flow (respectively, flow of weights) is trivial.(3) If (respectively, ) equals aZ, with a > 0, then the Krieger flow (respectively, flow of weights) is given by R R/aZ.https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press 20 T. Berendschot Suppose that M ⊂ B(H) is a von Neumann algebra represented on the Hilbert space H and that α : G M is a continuous action on M of an lcsc group G. Then the crossed product von Neumann algebra M α G ⊂ B(L 2 (G, H)) is the von Neumann algebra generated by the operators {π(x)} x∈M and {u h } h∈G acting on ξ ∈ L 2 (G, H) as https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press Phase transitions for non-singular Bernoulli actions 7 Assume that G (X, μ 1 ) = g∈G (X 0 , μ g ) is non-singular.For every t ∈ [0, 1] we have that μ e , for each t < 1 the Bernoulli action G (X, μ t ) = g∈G (X 0 , (1 − t)η e + tμ g ) is constructed in the same way, by starting with the conservative Bernoulli action G g∈G (C 0 , (1 − t)η e + tη g ).So for every t ∈ (t ) given by (3.2).Assume, moreover, that: (1) ν ∼ μ e , or (2) ν ≺ μ e and sup g∈G |log dμ g /dμ e (x)| < +∞ for a.e.x ∈ X 0 .Then there exists a t 0 > 0 such that G (X, μ t ) is strongly ergodic for every t < t 0 .https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press Phase transitions for non-singular Bernoulli actions 11 Proof of Theorem 3.1.h∈G H 2 (μ h , μ gh ) for every g ∈ G, Aut(T ) be a subgroup.When g ∈ G and e ∈ E, the edges e and g • e are simultaneously oriented towards, or away from ρ, unless e ∈ E([ρ, g • ρ]).If we start with a different root ρ ∈ T , let (μ e ) e∈E denote the corresponding family of probability measures on X 0 .Then we have that μ e = μ e for all but finitely many e ∈ E, so that the measures e∈E μ e and e∈E μ e are equivalent.Therefore, up to conjugacy, the action (4.2) is independent of the choice of root ρ ∈ T .Let T be a locally finite tree such that each vertex v ∈ V (T ) has degree at least 2. Suppose that G ⊂ Aut(T ) is a countable subgroup.Let μ 0 and μ 1 be equivalent probability measures on a standard Borel space X 0 and fix a root ρ ∈ T .Then the action α : G (X, μ) given by (4.2) is essentially free.Proof.Take g ∈ G \ {e}.It suffices to show that μ({x ∈ X : g https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press Let the free group F d on d ≥ 2 generators act on its Cayley tree T. Let μ 0 and μ 1 be equivalent probability measures on a standard Borel space X 0 . Then the action (4.2) dissipative if 1 − H 2 (μ 0 , μ 1 ) ≤ (2d − 1) −1/2 and weakly mixing and non-amenable if 1 − H 2 (μ 0 , μ 1 ) > (2d − 1) −1/2 .Furthermore, the action (4.2) is strongly ergodic when 1 − H 2 (μ 0 , μ 1 ) > (2d − 1) −1/4 .e ) if e is oriented towards ρ, log(dμ 0 /dμ 1 )(x e ) if e is oriented away from ρ.(4.3) Phase transitions for non-singular Bernoulli actions 25 LEMMA 4.5.Let T be a locally finite tree and let G ⊂ Aut(T ) be a closed subgroup.Suppose that H ⊂ G is a closed compactly generated subgroup that contains a hyperbolic element and assume that G/H is not compact.Let S ⊂ T be the unique minimal H-invariant subtree.Then there exists a vertex v ∈ S such that so that the flow of weights of G X × Y is as stated in the theorem.https://doi.org/10.1017/etds.2023.24Published online by Cambridge University Press