A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms

Let M be a compact manifold of dimension at least 2, Diff^r(M) be the space of C^r diffeomorphisms of M. Define for any diffeomorphism f in Diff^r(M) number of isolated periodic points of period n by P_n(f)=# {isolated x in M: f^n(x)=x}. Artin--Mazur proved that for a dense set of diffeomorphisms the number of periodic points P_n(f) growth at most exponentially fast in n. The author proved that there is an open set N \subset Diff^r(M) such that for a Baire generic set of diffeomorphisms the number of periodic points P_n(f) growth arbitrarily fast. Arnold posed a problem: Prove that diffeomorphisms with at most exponential growth of the number of periodic points in period have probability one. In this paper we annonce and exhibit key ingredients for a partial solution to Arnold's problem: we prove that for any epsilon>0 with probability one the number of periodic points is bounded by a streched exponential estimate P_n(f)<exp(Cn^{1+epsilon}).