Tuesday Seminar on Topology

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Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

2015/05/12

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masayuki Asaoka (Kyoto University)
Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)
[ Abstract ]
For any hyperbolic dynamical system, the number of periodic
points grows at most exponentially and the growth rate
reflects statistic property of the system. For dynamics far
from hyperbolicity, the situation is different. In 1999,
Kaloshin proved genericity of super-exponential growth in the
region where dense set of dynamical systems exhibits homoclinic
tangency (so called the Newhouse region).

How does the number of periodic points grow for generic
partially hyperbolic dynamical systems? Such systems are known
to be far from homoclinic tangency. Is the growth at most
exponential like hyperbolic system, or super-exponential by
a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved
super-exponential growth of the number of periodic points for
generic one-dimensional iterated function systems under some
reasonable conditions. Such systems are models of dynamics
of partially hyperbolic systems in neutral direction. So, we
expect genericity of super-exponential growth in a region of
partially hyperbolic systems.

In this talk, we start with a brief history of the problem on
growth rate of the number of periodic point and discuss two
mechanisms which lead to genericity of super-exponential growth,
Kaloshin's one and ours.