p-Adic dynamical systems of Chebyshev polynomials |
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Authors: | B Diarra D Sylla |
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Institution: | 1. Laboratoire de Mathématiques, UMR 6620, CNRS-UBP, Complexe Scientifique des Cézeaux, 63171, Aubière, France 2. Faculté des Sciences et Techniques, DER de Mathématiques et Informatique, Univerisité des Sciences des Techniques et des Technologies de Bamako, BP: E 3206, Bamako, Mali
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Abstract: | We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ? p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w. |
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