EVENT DETAILS AND ABSTRACT

Title: Thermodynamic formalism of fiber-bunched $GL(d,R)$-cocycles
Speaker: Kiho Park
Speaker Info: University of Chicago
Brief Description:
Special Note:
Abstract:

We study subadditive thermodynamic formalism of H\”older and fiber-bunched $GL(d,R)$-cocycles over subshift of finite types. Here, fiber-bunched cocycles refer to cocycles that are nearly conformal. Unlike additive thermodynamic formalism where any H\”older continuous potential has a unique equilibrium state, there are examples of H\”older continuous matrix cocycles with multiple equilibrium states. Restricted to fiber-bunched cocycles, we show that there exists an open and dense subset of cocycles with unique equilibrium states; such open and dense subset consists of typical cocycles first introduced by Bonatti and Viana. The unique equilibrium states of typical cocycles follow from a property known as quasi-multiplicativity, and they have the subadditive Gibbs property.

When $d=2$, we have complete description of cocycles with unique equilibrium states. In particular, irreducible cocycles necessarily have unique equilibrium states, and we provide characterization for reducible cocycles with more than one equilibrium states.