Arithmetical method to detect integrability in maps.

We develop a method to detect the presence of integrals of the motion in symplectic rational maps, by representing these maps over finite fields and examining their orbit structure. We find markedly different orbit statistics depending upon whether the map is integrable or not.

Complexity of regular invertible p-adic motions.

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.