# Entropy, chaos and weak Horseshoe for infinite dimensional Random dynamical systems

Wednesday, October 24, 2012 - 10:15am - 11:05am

Keller 3-180

Wen Huang (University of Science and Technology of China)

In this talk, we present an answer to the long standing problem on the

implication of positive entropy of a random dynamical system. We study

C^0 infinite dimensional random dynamical systems in a Polish space, do not

assume any hyperbolicity, and prove that chaos and weak horseshoe exist

inside the random invariant set when its entropy is positive. This result

is new even for finite dimensional random dynamical systems and infinite

dimensional deterministic dynamical systems generated by either parabolic

PDEs or hyperbolic PDEs. We mention that in general one does not expect

to have a horseshoe without assuming hyperbolicity. For example, consider

the product system of a circle diffeomorphism with an irrational rotation

number and a system with positive entropy. This product system has positive

entropy and a weak horseshoe, but has no horseshoe.

This is a joint work with Prof. Kening Lu

implication of positive entropy of a random dynamical system. We study

C^0 infinite dimensional random dynamical systems in a Polish space, do not

assume any hyperbolicity, and prove that chaos and weak horseshoe exist

inside the random invariant set when its entropy is positive. This result

is new even for finite dimensional random dynamical systems and infinite

dimensional deterministic dynamical systems generated by either parabolic

PDEs or hyperbolic PDEs. We mention that in general one does not expect

to have a horseshoe without assuming hyperbolicity. For example, consider

the product system of a circle diffeomorphism with an irrational rotation

number and a system with positive entropy. This product system has positive

entropy and a weak horseshoe, but has no horseshoe.

This is a joint work with Prof. Kening Lu

MSC Code:

37Hxx

Keywords: