P-Adic dynamical systems of Chebyshev polynomials

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 ℚp, enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of Tp: they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚp, (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of Tp in the field ℚp, (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 Tn is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.