Complexity of regular invertible p-adic motions

Abstract

We consider issues of computational complexity that arise in the study of quasi-periodic motions (Siegel discs) over the p-adic integers, where p is a prime number. These systems generate regular invertible dynamics over the integers modulo p(k), for all k, and the main questions concern the computation of periods and orbit structure. For a specific family of polynomial maps, we identify conditions under which the cycle structure is determined solely by the number of Siegel discs and two integer parameters for each disc. We conjecture the minimal parametrization needed to achieve-for every odd prime p-a two-disc tessellation with maximal cycle length. We discuss the relevance of Cebotarev's density theorem to the probabilistic description of these dynamical systems. (c) 2001 American Institute of Physics.

The size of quadratic p -adic linearization disks

“…Remark 1.2. Our estimate of r(f ) in Theorem A extends results obtained for quadratic polynomials over Q p by Ben-Menahem [3], and Thiran, Verstegen, and Weyers [29], and for certain polynomials with maximal multipliers over the p-adic integers Z p by Pettigrew, Roberts and Vivaldi [24], as well as results on small divisors by Khrennikov [13].…”

“…Remark 1.2. Our estimate of r(f ) in Theorem A extends results obtained for quadratic polynomials over Q p by Ben-Menahem [3], and Thiran, Verstegen, and Weyers [29], and for certain polynomials with maximal multipliers over the p-adic integers Z p by Pettigrew, Roberts and Vivaldi [24], as well as results on small divisors by Khrennikov [13]. There is a number of results on the existence of critical points and/or failing of injectivity on the boundary of complex quadratic linearization disks since the work of Herman [10].…”

Linearization in ultrametric dynamics in fields of characteristic zero — Equal characteristic case

“…the maximal disc U , about the origin, such that the full conjugacy g • f • g −1 (x) = λx, holds for all x ∈ U . Estmates of linearization discs have appeard in several papers concerning the p-adic case [5,18,35,42] later generalized in [28], and in [29,31] concerning the prime characteristic case.…”

Estimates of linearization discs in $p$-adic dynamics with application to ergodicity

“…In particular, since λ is not a root of unity, f can have no periodic points on the linearization disc, except the fixed point x 0 . However, the semi-disc may contain other periodic points as well, as manifest in the papers [4,53]. In fact, the semi-disc is contained in the quasi-periodicity domain of f , defined as the interior of the set of points on the projective line P(C p ) = C p ∪ {∞} that are recurrent by f .…”