Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts
Jean-Pierre Conze Y. Guivarc'h
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4239-4269 doi: 10.3934/dcds.2013.33.4239
Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walks in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. A powerful tool in this study is the spectral gap property for the operator $P$ when it holds. We use it to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.
keywords: spectral gap recurrence local limit theorem non compact extension of dynamical system random walk random scenery. Nilmanifold
The CLT for rotated ergodic sums and related processes
Guy Cohen Jean-Pierre Conze
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 3981-4002 doi: 10.3934/dcds.2013.33.3981
Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system. The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta \in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f \circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that $(S_n^\theta f)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ \tau^n)$ is a regular process.
    Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.
keywords: K-systems. rotated process Central Limit Theorem
Central limit theorem for stationary products of toral automorphisms
Jean-Pierre Conze Stéphane Le Borgne Mikaël Roger
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1597-1626 doi: 10.3934/dcds.2012.32.1597
Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlòs and the Multiplicative Ergodic Theorem.
keywords: kicked stationary process multiplicative system Lyapunov exponents. Toral automorphisms central limit theorem

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