Weighted equilibrium states for factor maps between subshifts

2009/09/23 by De‐Jun Feng, Feng, De-Jun · 1 citation
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0909.4250

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 \ba=(a₁,a₂)∈ \R² with a₁>0 and a₂≥ 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 μ(f)+a₁h_μ(σ_X)+ a₂h_{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 a₁h_μ(σ_X)+ a₂h_{μ∘ π^-1}(σ_Y). This answers an open question raised by Gatzouras and Peres in \citeGaPe96. An extension is also given to high dimensional cases. As an application, we show the uniqueness of invariant measures with full Hausdorff dimension for certain affine invariant sets on the k-torus under a diagonal endomorphism.

Cited by

• Measures of the full Hausdorff dimension for a general Sierpiński carpet

Related

• Measures of maximal relative entropy
• Existence of a measurable saturated compensation function between subshifts and its applications
• Equilibrium states which are not Gibbs measure on hereditary subshifts
• Decomposition of infinite-to-one factor codes and uniqueness of relative\n equilibrium states
• Equilibrium measures for two-sided shift spaces via dimension theory