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

Let (X,T)(X,T) be a topological dynamical system and F={fn}n=1∞ be a sub-additive potential on C(X,R)C(X,R). Let UU be an open cover of XX. Then for any TT-invariant measure μμ, let F∗(μ)=limn→∞1n∫fndμ. The topological pressure for open covers UU is defined for sub-additive potentials. Then we have a variational principle: P(T,F,U)=supμ{hμ(T,U)+F∗(μ):μ∈M(X,T)} where hμ(T,U)hμ(T,U) denotes the measure-theoretic entropy of μμ relative to UU and the supremum can be attained by a TT-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)P(T,F)=limdiam(U)→0P(T,F;U). Moreover, we obtained P(T,F)=supμ{hμ(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].