SRB measures for partially hyperbolic systems

Speaker: 

Yakov Pesin

Institution: 

Penn State University

Time: 

Tuesday, April 28, 2015 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In this series of three lectures I will consider SRB measures (after Sinai, Ruelle and Bowen) which arguably form one of the most important classes of invariant measures with “chaotic” behavior in dynamics. This ensures a crucial role they play in applications of dynamical systems to science (this is why they are often called “physical measures”).

In the first lecture I introduce these measures and describe their ergodic properties. I also outline a construction of SRB measures for Anosov systems.

In the second lecture I consider the case of partially hyperbolic dynamical systems and outline a construction of SRB measures in this case. I also discuss an important role SRB measures play in the Pugh-Shub Stable Ergodicity problem (and I will also discuss this problem in a general setting).

The third lecture deals with the most general situation of the so-called “chaotic” attractors (or attractors with non-zero Lyapunov exponents), which appear in many models in physics, biology, etc. I will present a general rigorous definition of chaotic attractors and outline a construction to build SRB measures for this attractors.

While the first lecture will serve as an introduction to the subject and will be accessible for students, the other two lectures are more advanced.

A symbolic representation of Anosov-Katok Diffeomorphisms IV

Speaker: 

Matt Foreman

Institution: 

UC Irvine

Time: 

Tuesday, April 14, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

I present joint work with B. Weiss that describes a concrete operation on words that allows one to generate symbolic representations of Anosov-Katok diffeomorphisms. We show that each A-K diffeomorphism can be represented this way and that each symbolic system generated by this operation can be realized as an A-K diffeomorphism.

A symbolic representation of Anosov-Katok Diffeomorphisms III

Speaker: 

Matt Foreman

Institution: 

UC Irvine

Time: 

Tuesday, March 3, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

I present joint work with B. Weiss that describes a concrete operation on words that allows one to generate symbolic representations of Anosov-Katok diffeomorphisms. We show that each A-K diffeomorphism can be represented this way and that each symbolic system generated by this operation can be realized as an A-K diffeomorphism.

Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

Speaker: 

Maciej Malicki

Institution: 

Department of Mathematics and Mathematical Economics, Warsaw School of Economics

Time: 

Tuesday, March 10, 2015 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

Inspired by a recent work of Marcin Sabok, we define a criterionfor a homogeneous metric structure X that implies that its automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space, regarded as metric structures, thus proving that their automorphism groups share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group with a left-invariant, complete metric, is trivial, and we verify it for the Urysohn space, and the Hilbert space.

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