Un système dynamique lié à la suite des nombres premiers

We study the dynamical system defined by the map $\text{Φ:}\text{]0,1]→}\text{]0,1]}$, 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 $\text{P}$ on the union of orbits stemming from numbers 1/p (p prime). We prove that ergodicity of $\text{P}$ 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.