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

Abstract

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $<f;{\mathbf{S}}_{2^{-r}}(a)>$ 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 $f:{\mathbb{Z}}_2\to {\mathbb{Z}}_2$ 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 ${\mu }_2({\mathbb{Z}}_2)=1$.

References in corpus (3)

• T-functions revisited: New criteria for bijectivity/transitivity
• Automata finiteness criterion in terms of van der Put series of automata functions
• Ergodic Theory Over ${\F}_2[[T]]$

Cited by in corpus (3)

• $p$-adic dynamical systems of $(2,2)$-rational functions with unique fixed point
• Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding
• Dynamical systems of the $p$-adic $(2,2)$-rational functions with two fixed points