Variational principle of higher dimension weighted pressure for amenable group actions

Abstract

Let r≥2 and (Xi,G)(i=1,⋯,r) be topological dynamical systems with an infinite countable discrete amenable phase group G. Suppose that πi:(Xi,G)→(Xi+1,G) are factor maps, a={a1,⋯,ar−1}∈Rr−1 is a vector with 0≤ai≤1 and (w1,⋯,wr)∈Rr is a probability vector associated with a. In this paper, given f∈C(X1), we introduce the weighted topological pressure Pa(f,G). Moreover, by using measure-theoretical theory, we establish a variational principle:Pa(f,G)=supμ∈MG(X1)⁡(∑i=1rwihμi(Xi,G)+w1∫X1fdμ), where h{⋅}(⋅,G) is the Kolmogorov-Sinai entropy of the systems acted by the amenable group G and μi=πi−1∘⋯∘π1μ is the induced G-invariant measure on Xi.