[Paper Review] Weighted equilibrium states for factor maps between subshifts

De‐Jun Feng|ArXiv.org|Sep 23, 2009

Mathematical Dynamics and Fractals31 references3 citations

TL;DR

This paper establishes the existence and uniqueness of weighted equilibrium states for factor maps between subshifts under weak specification, generalizing classical equilibrium state theory. It proves that for a continuous function $ f $ on a subshift $ X $ with weak specification and a factor map $ \pi: X \to Y $, there is a unique invariant measure maximizing $ \mu(f) + a_1 h_\mu(\sigma_X) + a_2 h_{\mu \circ \pi^{-1}}(\sigma_Y) $, resolving an open question by Gatzouras and Peres on weighted entropy maximization.

ABSTRACT

Let $π:X o Y$ be a factor map, where $(X,σ_X)$ and $(Y,σ_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. 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 $μ(f)+a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $μ$ on $X$ that maximizes the weighted entropy $a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. This answers an open question raised by Gatzouras and Peres in \cite{GaPe96}. 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.

Motivation & Objective

• To resolve an open question by Gatzouras and Peres on the uniqueness of invariant measures maximizing weighted entropy $ h_\mu(\sigma_X) + \alpha h_{\mu \circ \pi^{-1}}(\sigma_Y) $ for factor maps between subshifts.
• To extend equilibrium state theory to weighted combinations of measure-theoretic entropies across multiple subshifts via factor maps.
• To establish conditions under which a unique invariant measure maximizes a functional combining a continuous potential and multiple entropy terms.
• To apply the results to the uniqueness of invariant measures with full Hausdorff dimension for affine invariant sets on the torus.

Proposed method

• Uses the class $ V(\sigma_{X_1}) $ of functions with bounded variation in partial sums to ensure regularity of the associated subadditive potential.
• Reduces the weighted equilibrium state problem to a subadditive potential framework via the construction of a function $ \phi $ on cylinder sets with $ \phi(I) = \sup_{x \in [I]} \exp(S_n f(x)) $.
• Applies a recursive induction argument on the chain of factor maps $ X_1 \to X_2 \to \cdots \to X_k $, proving existence and uniqueness of weighted equilibrium states at each level.
• Employs the higher block representation to reduce general factor maps to one-block maps, preserving weak specification and simplifying analysis.
• Uses the variational principle to relate the weighted topological pressure to the supremum of the weighted entropy functional.
• Establishes mixing-type lower bounds on measure recurrence via the weak specification property, ensuring uniqueness of the maximizing measure.

Experimental results

Research questions

• RQ1 Does a unique invariant measure exist that maximizes the weighted entropy $ a_1 h_\mu(\sigma_X) + a_2 h_{\mu \circ \pi^{-1}}(\sigma_Y) $ for a factor map $ \pi: X \to Y $ between subshifts with $ X $ satisfying weak specification?
• RQ2 Can the classical uniqueness result for equilibrium states (Bowen) be extended to weighted combinations of entropies across a factor map?
• RQ3 Is there a unique invariant measure on a subshift with weak specification that maximizes a functional combining a continuous potential and multiple entropy terms from factor systems?
• RQ4 Does the uniqueness of weighted equilibrium states imply the uniqueness of invariant measures with full Hausdorff dimension for self-affine sets on the torus?
• RQ5 Can the theory be extended to chains of $ k \geq 2 $ subshifts with factor maps between them?

Key findings

• For any $ f \in V(\sigma_{X_1}) $ and $ \mathbf{a} = (a_1, \dots, a_k) $ with $ a_1 > 0 $, $ a_i \geq 0 $, there exists a unique $ \mathbf{a} $-weighted equilibrium state $ \mu $ maximizing $ \mu(f) + \sum_{i=1}^k a_i h_{\mu \circ \tau_{i-1}^{-1}}(\sigma_{X_i}) $.
• The unique equilibrium state $ \mu $ is ergodic and satisfies a uniform lower bound on the recurrence of measurable sets: $ \liminf_{n \to \infty} \sum_{i=0}^p \mu(A \cap \sigma^{-n-i} B) \geq c \mu(A)\mu(B) $ for some $ c > 0 $.
• If $ X_1 $ satisfies specification (not just weak specification), then $ \mu $ satisfies the stronger mixing-type lower bound $ \liminf_{n \to \infty} \mu(A \cap \sigma^{-n} B) \geq c \mu(A)\mu(B) $.
• When $ f \equiv 0 $, the unique measure maximizing the weighted entropy $ \sum_{i=1}^k a_i h_{\mu \circ \tau_{i-1}^{-1}}(\sigma_{X_i}) $ exists, solving an open problem of Gatzouras and Peres.
• The results apply to self-affine Sierpinski sponges on the $ k $-torus under diagonal endomorphisms, ensuring uniqueness of invariant measures with full Hausdorff dimension.
• The theory extends to both one-sided and two-sided subshifts via the use of one-sided projections and measure transfer through the canonical coding maps $ \Gamma_i $.