`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 367 - 377, Volume 34, Issue 2, February 2014      doi:10.3934/dcds.2014.34.367

 
       Abstract        References        Full Text (392.8K)       Related Articles       

Vladimir Anashin - Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskiye Gory, 1-52, Moscow, 119991, GSP-1, Russian Federation (email)
Andrei Khrennikov - International Center for Mathematical Modeling, Linnæus University, S-35195 Växjö, Sweden (email)
Ekaterina Yurova - International Center for Mathematical Modeling, Linnæus University, S-35195 Växjö, Sweden (email)

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 $\mu_2(\mathbb Z_2)=1$.

Keywords:  Ergodic theory, 1-Lipschitz dynamics, 2-adic sphere, p-adic analysis.
Mathematics Subject Classification:  Primary: 37P20; Secondary: 11S82.

Received: October 2012;      Revised: May 2013;      Available Online: August 2013.

 References