Un système dynamique lié à la suite des nombres premiers

Abstract

We study the dynamical system defined by the map Φ: ]0,1]→ ]0,1] , where Φ( x)= px−1 on ]1/ p,1/ q] if q and p are consecutive prime numbers. We relate the existence of an absolutely continuous invariant measure to ergodicity of a Markov chain P on the union of orbits stemming from numbers 1/ p ( p prime). We prove that ergodicity of P implies ergodicity of Φ. We establish a link between transfer probabilities of order n and some sets of sequences of the symbolic dynamic. This leads to a way of computing these coefficients using Monte Carlo method. We propose an algorithm which leads to a density indicating a good experimental fit with a typical orbit.