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Ergodic Transformations in the Space of p-Adic Integers

2006
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AIP Conference Proceedings
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Let $\mathcal L_1$ be the set of all mappings $f\colon\Z_p\Z_p$ of the space of all $p$-adic integers $\Z_p$ into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping $f\in\mathcal L_1$ is ergodic with respect to the normalized Haar measure on $\Z_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\Z/p^k\Z$ modulo $p^k$, for all $k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings, are considered also. Results

doi:10.1063/1.2193107
fatcat:mham3tvkpjde3kpu6hh7rroxzm