Dynamics of the $p$-adic Shift and Applications

TL;DR

This paper introduces a new approach to modeling the Bernoulli shift on p-adic integers, demonstrating that small perturbations preserve Bernoulli properties and identifying polynomial maps that are isomorphic to the shift.

Contribution

It provides a novel realization of the Bernoulli shift in the p-adic setting and shows stability of this property under small perturbations, including polynomial maps.

Findings

Small perturbations of the p-adic Bernoulli shift remain Bernoulli.

Polynomial maps on _p are isomorphic to the Bernoulli shift.

The approach offers new insights into p-adic dynamical systems.

Abstract

We present a novel way of realizing the Bernoulli shift on $p$ symbols on the $p$-adic integers, where $p$ is a prime. By showing that suitably small perturbations of the shift are still Bernoulli we find many "nice" maps, such as polynomials on $\mathbb Z_p$, that are isomorphic to the (noninvertible) Bernoulli shift.