Variational principle for weighted topological pressure

Abstract

Let (X,T) and (Y,S) be two topological dynamical systems, and π:X→Y a factor map. Let a=(a1,a2)∈R2 with a1>0 and a2≥0, and f∈C(X). We define the a-weighted topological pressure of f, denoted by Pa(X,f), as an extension of the classical topological pressure. In this approach, we use the a-weighted Bowen balls to substitute the Bowen balls in the classical definition. We prove the following variational principle:Pa(X,f)=sup⁡{a1hμ(T)+a2hμ∘π−1(S)+∫fdμ}, 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 T2 under affine diagonal endomorphisms. A higher dimensional version of the result is also established.