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Abstract

In this thesis, we study subadditive thermodynamic formalism of fiber-bunched cocycles. In particular, we study the norm potentials and their equilibrium states of such cocycles. We present mainly three results, each of which can be viewed as a suitable generalization of the corresponding result for locally constant cocycles. The first result concerns with an open and dense subclass of fiber-bunched GLd(R)- cocycles called the typical cocycles. We show that the norm potentials of typical cocycles have unique equilibrium states with the subadditive Gibbs property. The main body of work amounts to establishing a property called quasi-multiplicativity, which implies the stated result. As a corollary, we obtain the continuity of the subadditive pressure and the equilibrium state (with respect to the weak-∗ distance) on such cocycles. Second, in joint work with Clark Butler, we analyze fiber-bunched GL2(R)-cocycles. Exploiting the low dimensionality of the cocycles, we fully analyze their norm potentials. We show that irreducible fiber-bunched GL2(R)-cocycles have unique equilibrium states. We also provide a criterion for their norm potentials to have multiple ergodic equilibrium states. Lastly, in joint work with Benjamin Call, we study the ergodic properties on these unique equilibrium states. Under the settings considered in the above two results, we show that if the unique equilibrium state is totally ergodic, then it has the Kolmogorov property. This is done by extending the result of Ledrappier.

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