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Exponential growth of number of periodic orbits is not generic
by
V. Kaloshin
Princeton University
documentclassarticle
Exponential Growth of number of periodic orbits is not
topologically generic
Consider the space of Ck diffeomorphisms of a compact
manifold M denoted by
Dk(M), k >= 2. 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
In 1965 Artin & Mazur showed that the set of A-M diffeomorphisms
is dense in Dk(M).
The first result is that the set of A-M diffeomorphisms
is not residual in Dk(M) with the uniform
Ck-topology, i.e. this set is not topologically generic.
This theorem has been conjectured by J.Mather. Moreover, we
proved that there is an open set N in the space
Dk(M) such that N contains a residual
set with arbitrary ahead given growth of number of periodic orbits.
Examples of particular dynamical systems with arbitrary quick
growth of number of periodic orbits have been presented by
Rozales-Gonsalez. But, a residual set in the segment [0, 1]
can have measure zero.
Let Bn be the unit ball.
So, the second result (joint with B.Hunt) says that
a generic n-parameter family of C\infty diffeomorphisms
{f\epsilon }\epsilon in Bn with not extremely fast growing
derivatives has the following property:
for almost all
\epsilon in Bn number of periodic points of f\epsilon
growing not extremety fast meaning that for some C=C(\epsilon) > 0
pn(f)=# {x in M:fn(x)=x } < exp (Cn).
Some related properties of generic diffeomorphisms
will be also discussed.
pn(f\epsilon) < exp (C n3 logn).
Date received: December 16, 1997
Copyright © 1997 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 # caas-03.