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May  2006, 15(2): 611-640. doi: 10.3934/dcds.2006.15.611

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

Vadim Kaloshin 1, and  Maria Saprykina 2,

1. 

Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91106, United States
2. 

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

Received  February 2005 Revised  December 2005 Published  March 2006

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: periodic points, homoclinic tangency., Orbit growth, measure-preserving transformations.
Mathematics Subject Classification: Primary: 37C35; Secondary: 34C25, 37C25, 28D05, 34C3.
Citation: Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611
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