Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits

Let M be a compact manifold of dimension three with a nondegenerate volume form Omega and Diff(Omega)(r) (M) be the space of C-r-smooth (Omega-) volume-preserving difffeomorphisms of M with 2 <= r <= infinity. In this paper we prove two results. One of them provides the existence of a Newhouse domain N in Diff(Omega)(r)(M). The proof is based on the theory of normal forms [13], construction of certain renormalization limits, and results from [23, 26, 28, 32]. To formulate the second one, associate to each diffeomorphism a sequence P-n(f) which gives for each n the number of isolated periodic points of f of period n. The main result of this paper states that for a Baire generic diffeomorphism f in N, the number of periodic points P-n(f) grows with n faster than any prescribed sequence of numbers {a(n)} (n is an element of Z+) along a subsequence, i.e., P-ni (f) > ani for some n(i) -> with infinity i -> infinity. The strategy of the proof is similar to the one of the corresponding 2-dimensional non volume-preserving result [16]. The latter one is, in its turn, based on the Gonchenko-Shilnikov-Turaev Theorem [8, 9].