Maximizing measures for expanding transformations

On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function whose maximizing measure is supported on a periodic orbit. We also show that maximizing measures can be obtained as weak limits of equilibrium states. We apply these theorems to the class of C(1+a) endomorphisms of the circle which are coverings of degree 2, uniformly expanding and orientation preserving. We prove that generically the invariant probability which maximizes the Lyapunov exponent is unique and that C(1+b) endomorphisms can be approximated in the C(1+a) topology by endomorphisms whose Lyapunov maximizing measure is supported on a periodic orbit.

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