Properties of periodic orbits of generic diffeomorphisms
by
Vadim Kaloshin
Princeton University
Coauthors: Brian Hunt

PROPERTIES OF PERIODIC ORBITS FOR GENERIC DIFFEOMORPHISMS Vadim Yu. Kaloshin Princeton University Consider the space of C^k diffeomorphisms of a compact manifold M denoted by D^k(M), k > 1. Call a diffeomorphism f A-M diffeomorphism if number of periodic points growth with at most exponential speed in n, i.e. for some C > 0 and all n

Pn(f)=# {x in M: fn(x)=x } < exp (Cn).

In 1965 Artin & Mazur showed that, if P_n(f) would denote number of isolated period n orbits, then the set of A-M diffeomorphisms is dense in D^k(M). Define a dynamical \zeta-function for an A-M diffeomorphism f by

\zetaf(z)= å n  znPn(f)/n.

In 1967 Smale posed a question: Is dynamical zeta function generically rational? THEOREM 1. The set of A-M diffeomorphisms is not residual in the space of C^k diffeomorphisms D^k(M) with the uniform C^k-topology, i.e. this set is not topologically generic. Therefore, Smale's question has the negative answer, because generically \zeta_f-function is not even analytic. Moreover, it is proved that there is an open set N in the space of diffeomorphisms D^k(M) such that N contains a residual set with ARBITRARY AHEAD GIVEN GROWTH of number of periodic orbits. The proof is based on a theorem of Gonchenko-Shilnikov-Turaev. Examples of particular dynamical systems with arbitrary quick growth of number of periodic orbits have been constructed by Rozales-Gonsalez. A period n point p of a diffeomorphism f:M --> M fⁿ(p)=p is called \gamma-hyperbolic if the minimal distance from eigenvalues of linearization dfⁿ(p) to the unit circle is \gamma. Define quantitative hyperbolicity of f by

\gamman(f)=minimum of \gamma-hyperbolicity of all period npoints of f.

THEOREM 2. (joint with B.Hunt) There is a dense set of diffeomorphisms D in D^k(M) such that for any f in D there is a constant C > 0 \gamma_n(f) >= exp(-C n³) for all n. The last Theorem is a generalization of a Yomdin's result, which has a worse estimate.

Date received: March 19, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabf-30.