Variational principle for weighted topological pressure

By De-Jun Feng, Wen Huang. 2014

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Let $\pi:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_\mu(T)+a_2h_{\mu\circ\pi^{-1}}(S)+\int f \;d\mu\right\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of classical topological pressure, but also provides a topological extension of dimension theory of invariant sets and measures on the torus under affine diagonal endomorphisms. A higher dimensional version of the result is also established.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/465c82dab9d6b7ed516dc5ab850691659a37c4bb