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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)
Software:
Maple
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References:
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