Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 611 - 640, Volume 15, Issue 2, June 2006      doi:10.3934/dcds.2006.15.611

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Vadim Kaloshin - Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91106, United States (email)
Maria Saprykina - Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada (email)

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].

Keywords:  Orbit growth, periodic points, measure-preserving transformations, homoclinic tangency.
Mathematics Subject Classification:  Primary: 37C35; Secondary: 34C25, 37C25, 28D05, 34C37.

Received: February 2005;      Revised: December 2005;      Available Online: March 2006.