The Local Topological Pressure Variational Principle For Sub-additive Potentials

Let (X,T) be a topological dynamical system. A sequence F={f_n}_n=1^∞of functions on X is called sub-additive if each f_n is continuous non-negative valued function on X such thatu is an open cover of X, for any T-invariant measureμ, let F_*(μ)=(?)f_ndμ. The topological pressure is defined for sub-additive potentials via open covers, then we set up a variational principleand the supremum can be attained by a T-invariant ergodic measure.where h_μ(T,u) denotes the measure-theoretic entropy ofμ, relative to u. The main purpose of this paper is to generalize the result of Wen Huang and Yingfei Yi [8], in the paper [8], they proved the local variational principles of pressure for additive potentials .Furthermore, if F satisfies lim sup lim sup (?) logη_?(T,ε)=0, whereη_n(F,ε)=sup supwe prove the result P(T,F)=(?)P(T,F;u).Then we have P(T,F)=(?) {h_μ(T)+F_*(μ):μ∈M(X,T)}. another proof of topological pressure variational principle for sub-additive potentials [4] can be abtained.