How to perturb Bernoulli shift map

Abstract

The Bernoulli shift map is a fundamental example of a chaotic map with applications in algorithm design, data analysis, and numerical simulation. When implementing the Bernoulli shift map in the binary system, some sorts of perturbation methods are employed to make its outputs have long periods for practical reasons. We here look at one of such methods that perturbs underlying state space, and apply modular arithmetic to analyze the behavior of periods attained by this method, which reveals a close relationship with Artin’s conjecture on primitive roots. As a consequence, we obtain an exhaustive list of values for a dominant parameter of this method that are best possible in a theoretical sense.