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Quasi-multiplicativity of Typical Cocycles

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We show that typical [in the sense of Bonatti and Viana (Ergod Theory Dyn Syst 24(5):1295–1330, 2004) and Avila and Viana (Port Math 64:311–376, 2007)] Hölder and fiber-bunched \(\text {GL}_d(\mathbb {R})\)-valued cocycles over subshifts of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.

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The author is very grateful to his advisor, Amie Wilkinson, for her support and numerous helpful discussions. The author would also like to thank Clark Butler for sharing his insights and for pointing out an error in Sect. 3 of the original draft, and Aaron Brown for many helpful suggestions. Lastly, the author also thanks De-Jun Feng for his comments, Ping Ngai Chung for improving the readability of the paper, and anonymous referees for many useful comments that helped improve the paper.

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Correspondence to Kiho Park.

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Park, K. Quasi-multiplicativity of Typical Cocycles. Commun. Math. Phys. (2020).

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