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A fibered system associated with the prime number sequence. (Sur un système fibré lié à la suite des nombres premiers.) (French) Zbl 1117.37301
Summary: We study the dynamical system defined by the transformation $$\Phi:\,]0,1]\leq ]0,1]$$ where $$\Phi(x)=px-1$$ if $$x\in ]1/p,1/q],$$ $$q$$ and $$p$$ being two consecutive prime numbers. The problem of the existence of an invariant absolutely continuous measure by $$\Phi$$ is related via a Markov chain argument to a conjecture concerning a set of prime number sequences. This hypothesis is corroborated by Monte Carlo simulations. We prove that this implies the statistical stability of the transformation $$\Phi$$ on the interval $$]0,2/3].$$ By using heuristic arguments, we define simplified versions of the Perron-Frobenius operator associated to $$\Phi.$$ Using Maple, we construct a probability density presenting a good experimental fit with the histograms of orbits stemming from fundamental constants.
##### MSC:
 37E05 Dynamical systems involving maps of the interval 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
##### Keywords:
Prime numbers; Markov chain
Maple
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##### References:
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