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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].
Mathematics Subject Classification: Primary: 37C35; Secondary: 34C25, 37C25, 28D05, 34C37.
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