Variational principle for weighted topological pressure

Abstract

Let $\pi :X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let $\mathbf{a}=(a_1,a_2)\in {\mathbb{R}}^2$ with $a_1>0$ and $a_2\ge 0$, and $f\in C(X)$. The $\mathbf{a}$-weighted topological pressure of $f$, denoted by $P^{\mathbf{a}}(X,f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$P^{\mathbf{a}}(X,f)=\sup \left\{a_1{h}_{\mu }(T)+a_2{h}_{\mu \circ {\pi }^{-1}}(S)+\int fd\mu \right\},$$ 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.

References in corpus (1)

• On the topological pressure of random bundle transformations in sub-additive case