Related papers: Mixing on Rank-One Transformations

We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic…

The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular…

We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.

The paper is devoted to equipartition of measured information for finite state processes over regular trees whose laws are invariant under all parity preserving tree automorphisms. We show almost everywhere equipartition for ergodic…

We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We…

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots…

From a dynamical viewpoint, basic phase transitions of statistical mechanics can be regarded as a breaking of ergodicity. While many random models exhibiting such transitions at the thermodynamics limit exist, finite-dimensional examples…

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or…

Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$ compact. We characterize spread-out probability measures on $G$ that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral…

A rank-one infinite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that for each $t\ne 0$, the Cartesian powers of the transformation $T_t$ are all ergodic.

We consider the time dependent probability distribution of a coarse grained observable Y whose evolution is governed by a discrete time map. If the map is mixing, the time dependent one-step transition probabilities converge in the long…

It has been shown that in one dimension the environment viewed by the particle process (EVP process) in quasi periodic random environment is uniquely ergodic and mixing under mild additional assumptions. Here we construct an analytic quasi…

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and…

In this paper some sufficient conditions are given for when two bounded rank-one transformations are isomorphic or disjoint. For commensurate, canonically bounded rank-one transformations, isomorphism and disjointness are completely…

We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X, T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition \[…

We develop operator renewal theory for flows and apply this to infinite ergodic theory. In particular we obtain results on mixing for a large class of infinite measure semiflows. Examples of systems covered by our results include…

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly…

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq…

We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.

Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that "most" functions are optimized by measures supported on a periodic orbit, and it has…