Entropy and Dimension Families Associated with Equilibrium Measures for Hyperbolic Dynamical Systems

Abstract

For a (one or two-sided) subshift of finite type 0 and a Holder continuous function ψ we consider the equilibrium measures {μ,β} βȦ0 corresponding to the Holder continuous functions φβ(χ) = -βψ, β ≥ 0. We provide a complete description of the entropy family FE(β) =hμβ(σ) and the dimension family Fd((β) = dim H μ,β p,tJ associated with these measures. Similar results are obtained for entropy and dimension families generated by equilibrium measures for (conformal) expanding maps and (conformal) Axiom A diffeomorphisms. As a consequence we show that for a typical Holder continuous function ψ the set {hμ,β(σ) β ≥ 0} contains all positive values of metric entropy.