Geometry, Topology and Dynamics Seminar
Alex Furman
UIC
Simplicity of the Lyapunov spectrum for some systems
Abstract: Let $(X,m,T)$ be an ergodic probability measure preserving system, and $F:X\to {\rm SL}_d(\mathbf{R})$ be a measurable integrable function.
The associated Lyapunov exponents $\lambda_1\ge \lambda_2\ge \dots\ge \lambda_d$ describe the asymptotic behavior of the products $F_n(x)=F(T^{n-1}x)\cdots F(Tx)F(x)$;
namely the exponential expansion/contraction rates under $F_n(x)$.
An important problem is to determine whether the exponents are all distinct: $\lambda_i>\lambda_{i+1}$.
Simplicity of the spectrum (in the above sense) for products of independent random variables was established by Guivarc'h-Raugi and Gol'dsheid-Margulis in 70s and 80s.
More recently, simplicity of the spectrum for a particular system related to Teichmuller flow and Interval Exchange Transformations was conjectured by Kontsevich-Zorich
and proved by Avila-Viana.
Another case of simplicity of the spectrum appeared in the work of Eskin-Mirzakhani (proved in Eskin-Matheus).
In the talk I will describe a joint work with Uri Bader in which we give a "soft" proof of simplicity of the Lyapunov spectrum for a class of systems, including the above mentioned
situations.
Monday October 28, 2013 at 3:00 PM in SEO 636