Ergodic Theory/Probability Seminar- Kiho Park

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Ergodic Theory/Probability Seminar
November 21, 2019
3:00PM - 4:00PM
Location
Math Tower 154

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Add to Calendar 2019-11-21 15:00:00 2019-11-21 16:00:00 Ergodic Theory/Probability Seminar- Kiho Park Title: Thermodynamic formalism of fiber-bunched GL(d,R)-cocycles   Speaker: Kiho Park - University of Chicago   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. Math Tower 154 Department of Mathematics math@osu.edu America/New_York public
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Title: Thermodynamic formalism of fiber-bunched GL(d,R)-cocycles
 
Speaker: Kiho Park - University of Chicago
 
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.

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