Critical orbits and attracting cycles in p-adic dynamics

Details Critical orbits and attracting cycles in p-adic dynamics Critical orbits and attracting cycles in p-adic dynamics

2012-01-08

arxiv.org/abs/1201.1605v1

Mathematics

Number Theory

Scientific paper

Let f be a rational function of degree d, at least 2, defined over a non-archimedean field of characteristic 0 or larger than d and residue characteristic p. We show that there exists a constant \epsilon, depending on p and d, such that if z is an unramified fixed point of f with f'(z)<\epsilon, then there is a critical point w that is strictly attracted to z, that is, a critical point with an infinite forward orbit accumulating at z. Moreover, \epsilon = 1 for p > d. Thus in the non-archimedean setting every sufficiently attracting fixed point attracts a critical point. As a corollary, we deduce that there are only finitely many conjugacy classes of non-Lattes post-critically finite rational functions of any given degree d defined over any given number field. We also use this to show that over the field C_p, any rational function of degree d

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