Paramétrisation du problème 3x+1 sur la droite réelle : diagramme de phase, renormalisation et intervalles errants

Abstract : The 3x + 1 problem, defined on the positive integers, deals with the iterations of the function T(x) = (3x + 1)/2 if x is odd and T(x) = x/2 otherwise. An analytical extension on real numbers has been studied by Chamberland, and later by the authors. We generalize this study on a class of real functions that depend on four parameters, so as to encompass many variants of the 3x + 1 problem. Following a heuristic approach, we obtain a two-dimensional "phase diagram'' characterizing the average asymptotic dynamics (i.e., for almost every real number close to ±∞). In a large part of the parameter space, we establish the average growth rate of real trajectories close to infinity, under a condition of uniform distribution modulo 2. Finally, we describe the intrinsic and extrinsic dynamics close to integers using a renormalization method. This leads us to distinguish four types of wandering intervals.
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Submitted on : Thursday, January 2, 2020 - 12:10:48 PM
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Nik Lygeros, Olivier Rozier. Paramétrisation du problème 3x+1 sur la droite réelle : diagramme de phase, renormalisation et intervalles errants. 2020. ⟨hal-02412727v2⟩

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