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

Abstract

Let ( X , T ) be a topological dynamical system and F = { f n } n = 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 n ∫ f n d μ . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle: P ( T , F , U ) = sup μ { h μ ( T , U ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } where h μ ( 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 ) = lim d i a m ( U ) → 0 P ( T , F ; U ) . Moreover, we obtained P ( T , F ) = sup μ { h μ ( 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].