In this paper we study the ergodic theory of a class of symbolic dynamical systems $(Ø, T, μ)$ where $T:Ø\to Ø$ the left shift transformation on $Ø=\prod_0^\infty\{0,1\}$ and $μ$ is a $\s$-finite $T$-invariant measure having the property that one can find a real number $d$ so that $μ(τ^d)=\infty$ but $μ(τ^{d-ε})<\infty$ for all $ε>0$, where $τ$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $μ$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d>0$ then $μ$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $μ$ is an infinite measure with scaling rate of order $n^d$. Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at $z=1$ is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.
We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-space-valued additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides in particular a sufficient condition for unique ergodicity.
We study the dynamics of Blaschke products in two dimensions, particularly the rates of growth of the degrees of iterates and the corresponding implications for the ergodic properties of the map.
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.
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…
The ergodic decomposition of a family of Hua-Pickrell measures on the space of infinite Hermitian matrices is studied. Firstly, we show that the ergodic components of Hua-Pickrell probability measures have no Gaussian factors, this extends a result of Alexei Borodin and Grigori Olshanski. Secondly, we show that the sequence of asymptotic eigenvalues of Hua-Pickrell random matrices is balanced in certain sense and has a "principal value" coincides with the $γ_1$ parameter of ergodic components. This allow us to complete the program of Borodin and Olshanski on the description of the ergodic deco…
We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterise ergodicity geometrically in terms of pattern frequencies. The general framework allows to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attentio…
Given a finite-to-one factor code $π: X \to Y$ between irreducible sofic shifts and an ergodic $ν$ on $Y$ with full support, it is known that the fiber $π^{-1}_*(ν)$ has at most $d_π$ ergodic measures in it where $d_π$ is the degree of $π$. We introduce the notion of multiplicity for ergodic measures on $X$ (that depends on $π$) and we prove that $d_π$ is the sum of the multiplicity of $μ$ where $μ$ runs over the ergodic measures in $π^{-1}_*(ν)$. We also build an appropriate generalization to infinite-to-one factor codes in relation to class degree and relatively maximal measures. We also def…
We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost additive, with finite range dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem, more precisely, that averages along a Foelner sequence converge uniformly to a limiting function. Moreover we give explicit error estimates for the approximation in the sup norm.
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the inve…
A class of fast-slow Hamiltonian systems with potential $U_\varepsilon$ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter $\varepsilon$ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for $\varepsilon\to0$ to a homogenised Hamiltonian system. We study the situation where $\varepsilon$ is small but positive. First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that …
Given a factor code $π$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $π$ is finite-to-one there is an invariant called the degree of $π$ which is defined the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $ν$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $ν$ and have maximal entropy among all measures in the fibre $π^{-1}\{ν\}$. We show t…
We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and $T_1,...,T_\ell$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, t…
In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal equicontinuous factor. This result is obtained as an application of a general criteria which states that if a minimal system is an almost finite to one extension of its maximal equicontinuous factor and has no infinite independent sets of length $k$ for some $k\ge 2$, then it has only finitely many ergodic measures.
We study polygonal billiards with reflection laws contracting the reflected angle towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic SRB measures whose basins cover a set of full Lebesgue measure.
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 $T$.
In this paper we show that the natural action of the symmetric group acting on the product space $\{0, 1 \}^{\mathbb{N}} $ endowed with a symmetric measure is approximately transitive. We also extend the result to a larger class of probability measures.
A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\it inverse limits} of families of finite partitions.
We interpret the Pascal-adic transformation as a generalized induced automorphism (over odometer) and formulate the $σ$-finite analog of odometer which is also known as "Hajian-Kakutani transformation" (former "Ohio state example"). We shortly suggest a sketch of the theory of random walks on the groups on the base of $σ$-finite ergodic theory.
We prove that a shift ergodic measure on a topologically mixing sub-shift is isomorphic to a Bernoulli shift whenever it is quasi invariant under permutations of finite number of coordinates. We prove also that Gibbs measures on topologically mixing subshift of finite type are quasi invariant.
The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of "corners" of finite size, must be finite. A similar result, Theorem 1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results, combined with the ergodic decomposition theorem of [3], imply that the infinite Hua-Pickrell measures of Borodin and Olshanski [2] have finite ergodic components…
Given a factor code $π$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $ν$ on $Y$, and a function $V$ on $X$ with summable variation, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $ν$ and maximize $h(μ) + \int V dμ$ among all measures in the fiber $π^{-1}(ν)$. If $ν$ is fully supported, this bound is the class degree of $π$. This generalizes a previous result for the special case of $V=0$.
Let $K$ be a compact metrizable group and $\Ga$ be a group of automorphisms of $K$. We first show that each $\ap \in \Ga$ is distal on $K$ implies $\Ga$ itself is distal on $K$, a local to global correspondence provided $\Ga$ is a generalized $\ov{FC}$-group or $K$ is a connected finite-dimensional group. We show that $\Ga$ contains an ergodic automorphism when $\Ga$ is nilpotent and ergodic on a connected finite-dimensional compact abelian group $K$.
The third part of the paper concludes the proof of the main result --- the description of the ergodic decomposition of infinite Pickrell measures. First it is shown that the scaling limit of radial parts of finite-dimensional infinite Pickrell measures is precisely the infinite Bessel point process. It is then established that the "gaussian parameter" almost surely vanishes for our ergodic components, and the convergence to the scaling limit is then established in the space of finite measures on the space of finite measures. Finally, singularity is established for Pickrell measures correspondi…
We show that the measure of the spectrum of Schrödinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a quantitative upper bound for the measure of the spectrum. This follows from a result we proved for ergodic Schrödinger operators with finitely valued potentials under two conditions on the recurrence property of the shift. We also show that one of these conditions is necessary for such result.
The trivial proof of the ergodic theorem for a finite set $Y$ and a permutation $T:Y\to Y$ shows that for an arbitrary function $f:Y\to{\mathbb R}$ the sequence of ergodic means $A_n(f,T)$ stabilizes for $n \gg |T|$. We show that if $|Y|$ is very large and $|f(y)| \ll |Y|$ for almost all $y$, then $A_n(f,T)$ stabilizes for significantly long segments of very large numbers $n$ that are, however, $\ll |T|$. This statement has a natural rigorous formulation in the setting of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard fo…
The objective of this note is to present the results from the two recent papers. We study the Navier--Stokes equation on the two--dimensional torus when forced by a finite dimensional white Gaussian noise. We give conditions under which both the law of the solution at any time t>0, projected on a finite dimensional subspace, has a smooth density with respect to Lebesgue measure and the solution itself is ergodic. In particular, our results hold for specific choices of four dimensional white Gaussian noise. Under additional assumptions, we show that the preceding density is everywhere strictly …
We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behavior and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described …
Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d\times d$ matrices and $\varSigma_{K}^+$ the compact space of all one-sided infinite sequences $i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$. An ergodic probability $μ_*$ of the Markov shift $θ\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$, is called "extremal" for $\bS$, if $ρ(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$ holds for $μ_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $ρ(\bS)$ denotes the generalized/joint spectral radius of $\bS$. Using extremal norm and Kingman subadditive …
We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.
Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite orthogonal group and infinite unitary group respectively. The classifications of ergodic probability measures with respect to the natural group action of $O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{C})$ are due to Olshanski. The original proofs for these results are based on the asymptotic representation theory. In this note, by applying the Vershik-Kerov method, we propose a simple method for obtaining these two classi…
We investigate ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite measure preserving ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.
A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated er…
Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $α\colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$α$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $α$ itself is cohomologous to a homomorphism (modulo a compact group).
Let $M=X/Γ$ be a geometrically finite negatively curved manifold with fundamental group $Γ$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{γ\inΓ |\ d(o,γ.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise…
For a Bratteli diagram $B$, we study the simplex $\mathcal{M}_1(B)$ of probability measures on the path space of $B$ which are invariant with respect to the tail equivalence relation. Equivalently, $\mathcal{M}_1(B)$ is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from $\mathcal{M}_1(B)$ and the structure and properties of the diagram $B$. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex $\mathcal{M}_1(B)$ is a singleton…
The main goal in this paper is to describe the geometric structure of invariant graphs of a certain class of skew products. Our focus is on attracting multi-graphs. An invariant multi-graph is an invariant compact set which is a finite union of invariant graphs, and thus consists of a finite number of points on each fiber. We introduce invariant bony multi-graphs and construct an open set of skew products over an invertible base map (solenoid map) having attracting invariant multi-graphs and bony multi-graphs which support finitely many ergodic SRB measures. In this study some thermodynamic pr…
We show that a dissipative, ergodic measure preserving transformation of a sigma-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these.
We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic.
We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $Γ$ \textless{} G be a discrete subgroup of G, and G = N AK the Iwasawa decomposition of G. If the geodesic flow on $Γ$\G admits a finite measure of maximal entropy, we prove that the action of N on $Γ$\G by right multiplication admits a unique invariant measure supported on points whose A-orbit does not diverge.
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…
Given a finite-to-one factor map $π: (X, T) \to (Y, S)$ between topological dynamical systems, we look into the pushforward map $π_*: M(X, T) \to M(Y,T)$ between sets of invariant measures. We investigate the structure of the measure fiber $π_*^{-1}(ν)$ for an arbitrary ergodic measure $ν$ on the factor system $Y$. We define the degree $d_{π,ν}$ of the factor map $π$ relative to $ν$ and the multiplicity of each ergodic measure $μ$ on $X$ that projects to $ν$, and show that the number of ergodic pre-images of $ν$ is $d_{π,ν}$ counting multiplicity. In other words, the degree $d_{π,ν}$ is the su…
We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.
The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure $m$ with minimal exponent $h$; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a $\sg$--finite invariant measure $μ$ …
We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic. We also give type $\text{III}_λ$ examples of such systems, $0<λ\leq 1$. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence $(n_i)…
Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given "performance" function. For a continuous self-map of a compact metric space and a dense set of continuous performance functions, we show that the existence of uncountably many ergodic maximizing measures. We also show that, for a topologically mixing subshift of finite type and a dense set of continuous functions there exist uncountably many ergodic maximizing measures which are fully supported and have positive entropy.
We study double averages along orbits for measure preserving actions of $\mathbb{A}^ω$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these averages, which in particular reproves their convergence in $L^p$ for any finite $p$ and for any choice of two $L^\infty$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.
Given a space $X$, a $σ$-algebra $\mathfrak{B}$ on $X$ and a measurable map $T:X \to X$, we say that a measure $μ$ is half-invariant if, for any $B \in \mathfrak{B}$, we have $μ(T^{-1}(B)\leq μ(B)$. In this note we present a generalization of Birkhoff's Ergodic theorem to $σ$-finite half-invariant measures.
An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.
We show that every totally ergodic generalised matrix equilibrium state is psi-mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.
We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for $k$-fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric …
