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

Abstract

Let $M$ be a compact manifold of dimension three with a non-degenerate volume form $\Omega$ and Diff$^r_\Omega(M)$ be the space of $C^r$-smooth ($\Omega$-) volume-preserving diffeomorphisms of $M$ with $2\le r< \infty$. In this paper we prove two results. One of them provides the existence of a Newhouse domain $\mathcal N$ in Diff$^r_\Omega(M)$. 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 $\mathcal N$, the number of periodic points $P_n(f)$ grows with $n$ faster than any prescribed sequence of numbers $\{a_n\}_{n \in \mathbb Z_+}$ along a subsequence, i.e., $P_{n_i}(f)>a_{n_i}$ for some $n_i\to \infty$ with $i\to \infty$. 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].