Ergodicity criteria for non-expanding transformations of 2-adic spheres

Vladimir Anashin; Andrei Khrennikov; Ekaterina Yurova

doi:10.3934/dcds.2014.34.367

Article Contents

2014, Volume 34, Issue 2: 367-377. Doi: 10.3934/dcds.2014.34.367

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Ergodicity criteria for non-expanding transformations of 2-adic spheres

1.
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskiye Gory, 1-52, Moscow, 119991, GSP-1
2.
International Center for Mathematical Modeling, Linnæus University, S-35195 Växjö

Received: October 2012

Revised: May 2013

Published online: August 2013

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Abstract

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.

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Mathematics Subject Classification: Primary: 37P20; Secondary: 11S82.

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