We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added condition (by P. Bálint, N. Chernov, D. Szász, and I. P. Tóth, in order to save this fundamental result) on the algebraic character of the smooth boundary components of the configuration space is unnecessary. Having saved the theorem in its original form by using additional ideas in the spirit of the initial proof, the result becomes stronger and it applies to a larger family of models.
The central limit theorem for Markov chains generated by iterated function systems consisting of orientation preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.
We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the condition of the so called ``Ansatz'' can be dropped. That condition assumed that almost every singular phase point had a hyperbolic trajectory after the singularity. Having this condition dropped, the cited theorem becomes much stronger and easier to apply. At the end of the paper two immediate corollaries of this improvement are discussed: One of them is the (fully hyperbolic) Bernoulli mixing property of every hard disk system (D=2), the other one claims…
In 1993, E. Lesigne proved a polynomial extension of the Wiener-Wintner ergodic theorem and asked two questions: does this result have a uniform counterpart and can an assumption of total ergodicity be replaced by ergodicity? The purpose of this article is to answer these questions, the first one positively and the second one negatively.
In a multi-agent system, unconditional (multiple) consensus is the property of reaching to (multiple) consensus irrespective of the instant and values at which states are initialized. For linear algorithms, occurrence of unconditional (multiple) consensus turns out to be equivalent to (class-) ergodicity of the transition chain (A_n). For a wide class of chains, chains with so-called balanced asymmetry property, necessary and sufficient conditions for ergodicity and class-ergodicity are derived. The results are employed to analyze the limiting behavior of agents' states in the JLM model, the K…
Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchine's recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried over to a non-commutative *-algebraic setting using the GNS-construction. This generalizes the corresponding measure theoretic results, in particular a variation of Khintchine's Theorem for ergodic systems, where the image of one set overlaps with another set, instead of with itself.
We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a generalization of Birkhoff's ergodic theorem for quasi invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.
In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.
Inspired by topological Wiener-Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.
For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.
We generalize an equidistribution theorem à la Bader-Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries of a simply connected connected Riemannian manifold with pinched negative curvature. We combine a functional analytic tool, namely the property RD of hyperbolic groups, together with a dynamical tool: an equidistribution theorem of Paulin, Pollicott and Schapira inspired by a result of Roblin. In particular, we deduce irreducibility of these new classes of bo…
We prove a nonuniformly hyperbolic version Livšic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, We also prove a $μ$-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system.
We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations where the Liouville measure is not (or not known to be) ergodic.
The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$…
We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and $α$-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a g…
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Linde…
The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Rie…
The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional bill…
