The local variational principle of topological pressure for sub-additive potentials

摘要

Let (X,T) be a topological dynamical system and F=fnn=1∞ be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let F*(μ)=lim_(n→∞) 1∫f_ndμ. The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle: P(T,F,U)=supμhμ(T,U)+F_*(μ):μ∈M(X,T) where hμ(T,U) denotes the measure-theoretic entropy of μ relative to U and the supremum can be attained by a T-invariant ergodic measure. The main purpose of this paper is to generalize a result of Huang and Yi (2007) [17]. In the paper [17], they proved the local variational principle of pressure for additive potentials. Furthermore, we prove the result P(T,F)=limdiam(U)→0P(T,F;U). Moreover, we obtained P(T,F)=supμhμ(T)+F_* (μ):μ∈M(X,T), which gives another proof of the topological pressure variational principle for sub-additive potentials from Cao et al. (2008) [14].