Equilibrium states for factor maps between subshifts

Abstract

Let π : X → Y be a factor map, where ( X , σ X ) and ( Y , σ Y ) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a = ( a 1 , a 2 ) ∈ R 2 with a 1 > 0 and a 2 ⩾ 0 . Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes ∫ f d μ + a 1 h μ ( σ X ) + a 2 h μ ∘ π − 1 ( σ Y ) . In particular, taking f ≡ 0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a 1 h μ ( σ X ) + a 2 h μ ∘ π − 1 ( σ Y ) , which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dim H μ = dim H K .