Orthogonal measures and ergodicity

Burgess-Mauldin have proven the Ramsey-theoretic result that continuous sequences \({\left( {{\mu _c}} \right)_{c \in {2^\mathbb{N}}}}\) of pairwise orthogonal Borel probability measures admit continuous orthogonal subsequences. We establish an analogous result for sequences indexed by 2^N/E⁰, the next Borel cardinal. As a corollary, we obtain a strengthening of the Harrington-Kechris-Louveau E₀ dichotomy for restrictions of measure equivalence. We then use this to characterize the family of countable Borel equivalence relations which are non-hyperfinite with respect to an ergodic Borel probability measure which is not strongly ergodic.     

Asymptotic theory of path spaces of graded graphs and its applications

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The main subject is the asymptotic structure of filtrations and a new notion of standardness. All graded graphs and all filtrations of Borel or measure spaces can be divided into two classes: the standard ones, which have a regular behavior at infinity, and the other ones. Depending on this property, the list of invariant measures can either be well parameterized or have no good parametrization at all. One of the main results is a general standardness criterion for filtrations. We consider some old and new examples which illustrate the usefulness of this point of view and the breadth of its applications.     

Interval exchange transformations and some special flows are not mixing

An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds. The author is partially supported by grant NSF MCS 78-15278.     

The automorphism group of the Gaussian measure cannot act pointwise

Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for BooleanG-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

Invariant Measure for Diffusions with Jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established. Accepted 24 February 1998     

Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

随机环境中的马氏链的不变测度与遍历性

本文考虑了一类特殊的随机环境的马氏链。假设随机“Doeblin”条件成立，我们证明了随机环境的马氏链的不变测度存在，且任何初始分布以指数收敛速度到些不变测度。进一步的，存在关于绕积算子遍历的不变测度。最后，我们得到了随机马氏链的强大数定律。     

Statistical properties of dynamical systems - Simulation and abstract computation

We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).     

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.     

An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

For a regular jointly measurable Markov semigroup on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this semigroup in terms of subsets of the state space. In this way we extend results by Costa and Dufour (J. Appl. Probab. 43:767?C781, 2006). As a consequence we obtain an integral decomposition of every invariant probability measure in terms of the ergodic probability measures. Our approach is completely centered around the reduction to and relationship with the case of a single regular Markov operator associated to the Markov semigroup, the resolvent operator, which enables us to fully exploit results in that situation (Worm and Hille in Ergod. Theory Dyn. Syst. 31(2):571?C597, 2011).     

Unique ergodicity of flows on the torus and the rotation number

We prove in this paper that any flow on the 2- torus with no singular points and periodic orbits which is generated by a vector fieldV=(P, Q) satisfyingV∈C ¹ orV∈C ⁰ andP≠0 is uniquely ergodic. Then we give an expression of the rotation number by using an invariant measure of a flow.     

Twist points of the von Koch snowflake

It is known that the set of twist points in the boundary of the von Koch snowflake domain has full harmonic measure. We provide a new, simple proof, based on the doubling property of the harmonic measure, and on the existence of an equivalent measure, invariant and ergodic with respect to the shift.

     

Geometry and dynamics of admissible metrics in measure spaces

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ?-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ?-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.     

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.     

Expansions in nearly linear stochastic dynamical problems

A class of stochastic differential equations is considered which arises by adding a nonlinear term with a small parameter δ in the drift coefficient of a linear stochastic system. First, for a fixed time an expansion in powers of δ of the expectations of functions is established. Second, under the assumptions guaranteeing the existence of a unique ergodic measure, the corresponding expansion of the expectations of functions with respect to the invariant measure in powers of δ, δ²,… is also established.     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

Is Lebesgue measure the only σ-finite invariant Borel measure?

S. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel measure on Rd is a constant multiple of Lebesgue measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel measure and by constant. According to a folklore result, if the measure is only defined for Borel sets, then the answer is affirmative. We show that if the measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel measure (in the wider sense) on Rd can be non-σ-finite when we restrict it to the Borel sets.     

On the distribution of points in projective space of bounded height

In this paper we consider the uniform distribution of points in compact metric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometries. In this setting we prove the analogue of Weyl's criterion and the Erdös-Turán inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of projective space over completions of number fields in some detail. An invariant measure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the analogous Erdös-Turán inequality, we prove that the set of all projective points over the number field with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.

     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.