Let $σ:\boldsymbolΣ\to\boldsymbolΣ$ be the left shift acting on $ \boldsymbolΣ $, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $σ$-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential $ A : \boldsymbolΣ \to \mathbb R $. Under certain conditions, we are able to show not only that $A$-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).
Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Ma…
We prove that if an ergodic action of a countable group on a probability space admits a generating partition having finite Shannon entropy then it admits a finite generating partition.
We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.
For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.
We show uniform convergence of Wiener-Wintner ergodic averages for ergodic actions of (not necessarily countable) locally compact, second countable, abelian (LCA) groups. As a by-product, we obtain a finitary version of the van der Corput inequality for such groups.
A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n)_{n \in F}$ of a countable field $F$ with characteristic zero on a probability space $(X,\mathcal{B},μ)$ and a family $\{p_1,\dots,p_k\}$ of independent polynomials, we have \[ \lim_{N \to \infty} \frac{1}{|Φ_N|}\sum_{n \in Φ_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod_{j=1}^k \int_…
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergod…
We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.
Let $Γ$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $Γ$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $Γ$.
We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by Kaimanovich-Vershik and Karlsson-Ledrappier. We then proceed to prove that the product of the Poisson boundary of any countable measured group $(G,μ)$ with any ergodic $(G,\checkμ)$-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of $(G,μ)$ when $μ$ is symmetric. Finally, we characterize the failure of wea…
We prove that for all ergodic extensions S_1 of a transformation by a locally compact second countable group G, and for all G-extensions S_2 of an aperiodic transformation, there is a relative speedup of S_1 that is relatively isomorphic to S_2. We apply this result to give necessary and sufficient conditions for two ergodic n-point or countable extensions to be related in this way.
Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the disco…
It is shown that for each $N>0$ and for a wide class of Abelian non-compact locally compact second countable groups $G$ including all infinite countable discrete ones and $\Bbb R^{d_1}\times\Bbb Z^{d_2}$ with $d_1,d_2\ge 0$, there exists a weakly mixing probability preserving $G$-action with a homogeneous spectrum of multiplicity $N$.
We study ergodic properties of certain piecewise smooth two-dimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correleations.
A measure preserving action of a countably infinite group Γis called totally ergodic if every infinite subgroup of Γacts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γis totally ergodic then there exists a finite normal subgroup N of Γsuch that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely o…
For an ergodic probability-measure-preserving action $G \curvearrowright (X, μ)$ of a countable group $G$, we define the Rokhlin entropy $h_G^{\mathrm{Rok}}(X, μ)$ to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if $h_G^{\mathrm{Rok}}(X…
The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X \times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x \to μ_x^G$ is continuous, where $μ_x^G$ is the unique ergodic measure on $\{\ol{Orb(x)}, G\}$.
We show that for any countable amenable group action, along Følner sequences that have for any $c>1$ a two sided $c$-tempered tail, one have universal estimate for the probability that there are $n$ fluctuations in the ergodic averages of $L^{\infty}$ functions, and this estimate gives exponential decay in $n$. Any two-sided Følner sequence can be thinned out to satisfy the above property, and in particular, any countable amenble group admits such a sequence. This extends results of S. Kalikow and B. Weiss for $\mathbb{Z}^{d}$ actions and of N. Moriakov for actions of groups with polynomial gr…
Consider a subshift over a finite alphabet, $X\subset Λ^{\mathbb Z}$ (or $X\subsetΛ^{\mathbb N_0}$). With each finite block $B\inΛ^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\inΛ^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $μ$ on $X$, whose restriction to the space of measures is compatible with the weak-$\star$ topology. Next, in this combined metric space we fix an open set $\mathcal U$ containing all ergodic measures, and we…
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin ent…
Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group. A list of related open problems is provided.
Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary representation of G associated with T. In the case when G is either a discrete countable Abelian group or R^n, n>0, it is shown that the sets of the form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.
In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of …
Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for n…
We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.
Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.
We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.
We show that the stabilization of any countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions.
We present a general new method for constructing pointwise ergodic sequences on countable groups, which is applicable to amenable as well as to non-amenable groups and treats both cases on an equal footing. The principle underlying the method is that both cases can be viewed as instances of the general ergodic theory of amenable equivalence relations.
An algebraic $Γ$-action is an action of a countable group $Γ$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $Γ$ on a connected group $X$ is ergodic. We also show that this result does not hold for actions of polycyclic groups.
Let $\Cal S$ be the equivalence relation induced by the action SL$_2(\Bbb Z)\curvearrowright (\Bbb T^2,λ^2)$, where $λ^2$ denotes the Haar measure on the 2-torus, $\Bbb T^2$. We prove that any ergodic subequivalence relation $\Cal R$ of $\Cal S$ is either hyperfinite or rigid in the sense of S. Popa ([Po06]). The proof uses an ergodic-theoretic criterion for rigidity of countable, ergodic, probability measure preserving equivalence relations. Moreover, we give a purely ergodic-theoretic formulation of rigidity for free, ergodic, probability measure preserving actions of countable groups.
In a previous article, we extended the notion of ergodic optimization to the setting of C*-dynamical systems of countable discrete groups. Among the key results of that paper was that given an action $G \stackrelΞ{\curvearrowright} \mathfrak{M}$ of a countable discrete amenable group $G$ on a W*-probability space $(\mathfrak{M}, ρ)$ by $ρ$-preserving $*$-automorphisms of $\mathfrak{M}$, a positive element $x \in \mathfrak{M}$, and a right Følner sequence $\mathcal{F} = (F_k)_{k \in \mathbb{N} }$ for $G$, the sequence $$\left( \left\| \frac{1}{|F_k|} \sum_{g \in F_k} Ξ_g x \right\| \right)_{ k …
We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers. We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Pl…
We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alp…
We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation…
It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable. If $G$ has the Haagerup property then (and only then) these actions can be chosen of 0-type. If $G$ is amenable and unimodular then $G$ has weakly mixing Bernoulli actions of any possible Krieger's type.
Given a transformation $T$ of a standard measure space $(X,μ)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,μ)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this survey paper which subsets of $\Bbb N\cup\{\infty\}$ are realizable as $\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.
We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors $N=L^{\infty}(X)\rtimesΓ$ associated to free ergodic probability measure preserving (p.m.p.) actions $Γ\curvearrowright (X,μ)$ such that either (a) $Γ$ is a free group, $\mathbb F_n$, for some $n\geq 2$, or (b) $Γ$ is a non-inner amenable group and the orbit equivalence relation of the action $Γ\curvearrowright (X,μ)$ satisfies a property introduced in \cite{JS85}. On the…
A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergod…
Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.
We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a highly uniform bound on the number of fluctuations of the ergodic average for a class of Følner sequences satisfying an analogue of Lindenstrauss's temperedness condition. Equivalently, we deduce a uniform bound on the number of fluctuations over long distances for arbitrary Følner sequences. As a corollary, these results imply associated bounds for a continuous…
The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimen…
A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergodic extensions of a fixed, but arbitrary, ergodic transformation $T_0$. We then use a result of Connes, Feldman and Weiss to extend this relative theorem to the general (countable) amenable group.
Let $η$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting an invariant probability measure $μ$, such that the resulting dynamical system $(X, μ, F_3)$ is strongly ergodic and distal of rank $η$. In particular this shows that there is a $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.
We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other di…
Let $Γ$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $Γ\curvearrowright [0,1]^Γ$, where $[0,1]^Γ$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\Cal R$ of $\Cal S$, there exists a partition $\{X_i\}_{i\geq 0}$ of $[0,1]^Γ$ with $\Cal R$-invariant measurable sets such that $\Cal R_{|X_0}$ is hyperfinite and $\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\geq 1$.
We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a highly uniform bound on the number of fluctuations of the ergodic average for a class of Følner sequences satisfying an analogue of Lindenstrauss's temperedness condition. Equivalently, we deduce a uniform bound on the number of fluctuations over long distances for arbitrary Følner sequences. As a corollary, these results imply associated bounds for a continuous…
