# Generic diffeomorphisms with superexponential growth of number of periodic orbits

Kaloshin V. 2000. Generic diffeomorphisms with superexponential growth of number of periodic orbits. Communications in Mathematical Physics. 211, 253–271.

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Abstract

Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be the space of C r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diffr(M) the sequence P n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points P n f grows with a period n faster than any following sequence of numbers {a n } n ∈ Z + along a subsequence, i.e. P n (f)>a ni for some n i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented.

Publishing Year

Date Published

2000-04-01

Journal Title

Communications in Mathematical Physics

Volume

211

Page

253-271

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### Cite this

Kaloshin V. Generic diffeomorphisms with superexponential growth of number of periodic orbits.

*Communications in Mathematical Physics*. 2000;211:253-271. doi:10.1007/s002200050811Kaloshin, V. (2000). Generic diffeomorphisms with superexponential growth of number of periodic orbits.

*Communications in Mathematical Physics*. Springer Nature. https://doi.org/10.1007/s002200050811Kaloshin, Vadim. “Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.”

*Communications in Mathematical Physics*. Springer Nature, 2000. https://doi.org/10.1007/s002200050811.V. Kaloshin, “Generic diffeomorphisms with superexponential growth of number of periodic orbits,”

*Communications in Mathematical Physics*, vol. 211. Springer Nature, pp. 253–271, 2000.Kaloshin, Vadim. “Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.”

*Communications in Mathematical Physics*, vol. 211, Springer Nature, 2000, pp. 253–71, doi:10.1007/s002200050811.