Exponential mixing implies Bernoulli
Let $f$ be a $C^{1+α}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $μ$. We show that if $f:(M,μ)\to (M,μ)$ is exponentially mixing then it is Bernoulli.
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arXiv subjects
Dmitry Dolgopyat
Publications and source records attributed to Dmitry Dolgopyat.
Infinite measure mixing for some mechanical systems
We show that if an infinite measure preserving system is well approximated on most of the phase space by a system satisfying the local limit theorem, then the original system enjoys mixing with respect to global observables, that is, the observables which admit an infinite volume average. The systems satisfying our conditions include the Lorentz gas with Coulomb potential, the Galton board and piecewise smooth Fermi-Ulam pingpongs.
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Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence
A classical Borel Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will almost surely happen. In this article, we propose an extension of Borel Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions, Diophantine approximations and extreme value theory for dynamical systems.
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Global observables for random walks: law of large numbers
We consider the sums $T_N=\sum_{n=1}^N F(S_n)$ where $S_n$ is a random walk on $\mathbb Z^d$ and $F:\mathbb Z^d\to \mathbb R$ is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that $T_N$ always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on $F$, we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.
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Edgeworth expansions for independent bounded integer valued random variables
We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the underlying partial sums modulo some $h\in \bbN$ and the uniform distribution could fail to be $o(σ_N^{1-r})$, where $σ_N$ is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order $r$ fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal.
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Mixing properties of generalized $T, T^{-1}$ transformations
We study mixing properties of generalized $T, T^{-1}$ transformation. We discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is sufficient that the driving cocycle is small with small probability. In the case the fiber dynamics is only assumed to be ergodic, one needs to use the shearing properties of the cocycle. Applications include the Central Limit Theorem for sufficiently fast mixing systems and the estimates on deviations of ergodic averages.
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Constructive approach to limit theorems for recurrent diffusive random walks on a strip
We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.
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Deviations of ergodic sums for toral translations II. Boxes
We study the Kronecker sequence $\{nα\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $α$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.
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Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment
We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.
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Limit theorems for toral translations
We discuss some classical and recent results and open problems on the statistical behavior of ergodic sums above toral translations, and their applications to Diophantine approximations and to ergodic properties of systems related to quasi-periodic dynamics such as skew products, cylindrical cascades and special flows.
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Flexibility of statistical properties for smooth systems satisfying the central limit theorem
In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties: (1) zero entropy; (2) weak but not strong mixing; (3) (polynomially) mixing but not $K$; (4) $K$ but not Bernoulli; (5) non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index $1$.
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Dispersing Fermi-Ulam Models
We study a natural class of Fermi-Ulam Models that features good hyperbolicity properties and that we call dispersing Fermi-Ulam models. Using tools inspired by the theory of hyperbolic billiards we prove, under very mild complexity assumptions, a Growth Lemma for our systems. This allows us to obtain ergodicity of dispersing Fermi-Ulam Models. It follows that almost every orbit of such systems is oscillatory.
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Local Limit Theorems for Random Walks in a Random Environment on a Strip
The paper consists of two parts. In the first part we review recent work on limit theorems for random walks in random environment (RWRE) on a strip with jumps to the nearest layers. In the second part, we prove the quenched Local Limit Theorem (LLT) for the position of the walk in the transient diffusive regime. This fills an important gap in the literature. We then obtain two corollaries of the quenched LLT. The first one is the annealed version of the LLT on a strip. The second one is the proof of the fact that the distribution of the environment viewed from the particle (EVFP) has a limit for a. e. environment. In the case of the random walk with jumps to nearest neighbours in dimension one, the latter result is a theorem of Lally \cite{L}. Since the strip model incorporates the walks with bounded jumps on a one-dimensional lattice, the second corollary also solves the long standing problem of extending Lalley's result to this case.
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Asymptotic expansion of correlation functions for $\mathbb{Z}^d$ covers of hyperbolic flows
We establish expansion of every order for the correlation function of sufficiently regular observables of $\mathbb Z^d$ extensions of some hyperbolic flows. Our examples include the $\mathbb Z^2$ periodic Lorentz gas and geodesic flows on abelian covers of compact manifolds with negative curvature.
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Multi-type branching processes with time dependent branching rates
Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to infinity.
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No temporal distributional limit theorem for a.e. irrational translation
Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero.
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On mixing and the local central limit theorem for hyperbolic flows
We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors.
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Infinite measure renewal theorem and related results
We present abstract conditions under which a special flow over a probability preserving map with a non-integrable roof function is Krickeberg mixing. Our main condition is some version of the local central limit theorem for the underlying map. We check our assumptions for iid random variables (renewal theorem with infinite mean) and for suspensions over Pomeau-Manneville maps.
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