Growth of the number of periodic points for generic diffeomorphisms


Vadim Kaloshin


Brandeis University

Thursday, October 4, 2001


Talk at 4:30 p.m. in 317 Goldsmith Hall

Tea at 4:00 p.m. in 300 Goldsmith Hall


Abstract:   Let M be a compact manifold, and f a Cr-smooth map from M to M. Define the number of isolated periodic points of period n by

Pn(f) = #{isolated points x in M such that fn(x)=x}.

Artin and Mazur (1965), using Nash's approximations, proved that for a dense set of Cr-smooth maps, the number of periodic points Pn(f) grows at most exponentially fast, i.e. for some C > 0

(*)    Pn(f) <= exp(C n), for any positive integer n.

For more than thirty years there was no other proof known. During the talk we shall give an elementary proof of this result due to the author.

Various questions about relation of property (*), the dynamical zeta-function (Smale 1967), and topological entropy (Bowen 1978), and other for generic diffeomorphisms were raised. It turns out that Artin-Mazur property (*) is not generic, which was shown by the author using a theorem of Gonchenko, Shilnikov, and Turaev. Therefore, all the above questions have negative answer. Other aspects of the problem will be discussed.

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