Deterministic Walks in Quenched Random Environments of Chaotic Maps

On-off intermittency and chaotic walks

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on-off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

On the quenched functional CLT in random sceneries

We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a Z d-random walk in different frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic flows on homogeneous spaces).

Dynamical properties of random walks

In this paper, we study dynamical properties as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreductible Markov chains. As particular cases, we consider simple random walks on Z and Z+.

Random Walk in deterministically changing environment

We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.

包括ABC流在内的一些典型三维流的随机分裂的淬火相关性衰减

Quenched correlation decay for random splittings of some prototypical 3D flows including the ABC flow

本文针对包括ABC流在内的典型三维流，采用随机分裂方法，证明了拉格朗日混沌和几乎必然淬火相关性衰减，并应用于构建理想发电机和建立被动标量的指数混合。