Measure-preserving Actions Research Articles

Character rigidity for lattices and commensurators

abstract: We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

Uniform syndeticity in multiple recurrence

Abstract The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,l\geq 1$ and any $\varepsilon> 0$ , we prove the existence of $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that the following holds: Consider a solvable group $\Gamma $ of derived length l, a probability space $(X, \mu )$ , and d pairwise commuting measure-preserving $\Gamma $ -actions $T_1, \ldots , T_d$ on $(X, \mu )$ . Let E be a measurable set in X with $\mu (E) \geq \varepsilon $ . Then, K many (left) translates of $$ \begin{align*} \big\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \cdots \circ T^{\gamma^{-1}}_1(E))\geq \delta \big\} \end{align*} $$ cover $\Gamma $ . This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\geq 1$ and any $\varepsilon> 0$ , there are $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that for all finite solvable groups G of derived length l and any subset $E\subset G^d$ with $m^{\otimes d}(E)\geq \varepsilon $ (where m is the uniform measure on G), we have that K-many (left) translates of $$ \begin{align*} \{g\in G\colon &m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon \\ & (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq \delta \} \end{align*} $$ cover G. The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.

Non-existence of a universal zero-entropy system via generic actions of almost complete growth

Abstract We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

Existentially closed measure-preserving actions of free groups

Existentially closed measure-preserving actions of free groups

Multiorders in amenable group actions

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure \nu on the collection \widetilde{\mathcal O} of linear orders of type \mathbb Z on G , invariant under the natural action of G on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving G -action (X,\mu,G) has a multiorder (\widetilde{\mathcal O},\nu,G) as a factor and has the same orbits as the \mathbb Z -action (X,\mu,S) , where S is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra \Sigma_{\widetilde{\mathcal O}} associated with the multiorder factor is invariant under S , which makes the corresponding \mathbb Z -action (\widetilde{\mathcal O},\nu,\widetilde{S}) a factor of (X,\mu,S) . We prove that the entropy of any G -process generated by a finite partition of X , conditional with respect to \Sigma_{\widetilde{\mathcal O}} , is preserved by the orbit equivalence with (X,\mu,S) . Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to h(\mu,T,\mathcal P)=H(\mu,\mathcal P|\mathcal P^-) known for \mathbb Z -actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra \Sigma , as soon as the “orbit change” is measurable with respect to \Sigma . In our variant, we replace the measurability assumption by a simpler one: \Sigma should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any G -process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of G .

Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits

We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host–Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host–Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 k\geq 0 the joint distribution’s marginals on affine subcubes of dimension k k are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

Convergence of spherical averages for actions of Fuchsian groups

We prove pointwise convergence of spherical averages for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen–Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Measurable Tilings by Abelian Group Actions

Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $F \odot A = X$ implies $F^{r} \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus ${\mathbb{T}}^{d}$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).

Zero entropy actions of amenable groups are not dominant

AbstractA probability measure-preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb {Z}$ , an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, that is, that zero entropy implies non-dominance.

Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics.•Seward and Tucker-Drob showed that every free Borel action Γ↷X of a countable group Γ admits an equivariant Borel map π:X→Y to a free subshift Y⊂2Γ. We give a new simple proof of this result.•We show that for a countable group Γ, Free(2Γ) is weakly contained, in the sense of Elek, in every free continuous action of Γ on a zero-dimensional Polish space. This fact is analogous to the theorem of Abért and Weiss for probability measure-preserving actions and has a number of consequences in continuous combinatorics. In particular, we deduce that a coloring problem admits a continuous solution on Free(2Γ) if and only if it can be solved on finite subgraphs of the Cayley graph of Γ by an efficient deterministic distributed algorithm (this fact was also proved independently and using different methods by Seward). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.

Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-preserving action over a standard boundary action.

The relative f-invariant and non-uniform random sofic approximations

AbstractThe f-invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f-invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f-invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.

Compact families and typical entropy invariants of measure-preserving actions

For a compact set of actions, an entropy of Kushnirenko type is chosen in such a way that it vanishes on this set but takes infinite values for the typical actions. As a consequence we find that typical measure-preserving transformations are not isomorphic to isometric rearrangements of a finite set of geometric figures.

On a measurable analogue of small topological full groups II

We pursue the study of L 1 full groups of graphings and of the closures of their derived groups, which we call derived L 1 full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived L 1 full group has finite topological rank. We further show that a graphing is amenable if and only if its L 1 full group is, and explain why various examples of (derived) L 1 full groups fit very well into Rosendal’s geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between L 1 full groups of amenable ergodic graphings must be a quasi-isometry for their respective L 1 metrics. We finally show that L 1 full groups of rank one transformations have topological rank 2.

The Shannon–McMillan–Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

Measurable Hall’s theorem for actions of abelian groups

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

Classification of backward filtrations and factor filtrations: examples from cellular automata

AbstractWe consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik’s theory, but we also study them from another point of view that takes into account the measure-preserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.

Relative entropy and the Pinsker product formula for sofic groups

We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if X is a compact group, and G is a sofic group with G\curvearrowright X by automorphisms, then the outer Pinsker factor of G\curvearrowright (X,m_{X}) is given as a quotient by a G -invariant, closed, normal subgroup of X . We use our results to show that if G is sofic and f\in M_{n}(\mathbb Z(G)) is invertible as a convolution operator \ell^{2}(G)^{\oplus n}\to \ell^{2}(G)^{\oplus n}, then the action of G on the Pontryagin dual of \mathbb Z(G)^{\oplus n}/\mathbb Z(G)^{\oplus n}f has completely positive measure-theoretic entropy with respect to the Haar measure.