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Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2006 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, Vol. 15, no 2, 611-640 p.Article in journal (Refereed) Published
Abstract [en]

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].

Place, publisher, year, edition, pages
2006. Vol. 15, no 2, 611-640 p.
Keyword [en]
orbit growth, periodic points, measure-preserving transformations, homoclinic tangency, homoclinic tangencies, strange attractors, dynamics, points, hyperbolicity, bifurcations, horseshoes, dimension
URN: urn:nbn:se:kth:diva-15467ISI: 000235664000013OAI: diva2:333508
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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Saprykina, Maria.
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