Topological Tail Pressures And Variational Principles For Two Non-additive Potentials

In the paper, we discuss the topological tail pressure for sub-additive upper semi-continuous potentials and asymptotically sub-additive continuous potentials respectively on topological dynamical system, and present variational principles for the topological tail pressure without any additional assumptions.Firstly, we define the topological tail pressure via separated subsets and conditional pressure with respect to refining sequence of partitions respectively, then we prove the two pressures are equivalent under a refining sequence of essential partitions. Secondly, we discuss the properties of topological tail pressure, including the power rule, the product rule and the natural extension. Thirdly, according to the equivalence of the topological tail pressure and the conditional pressure, the properties of tail entropy function, we obtain the variational principle for sub-additive upper semi-continuous potentials:Let(X,T) be a topological dynamical system with finite topological entropy and F be a sub-additive upper semi-continuous potentials. Then P*(T,F)＝sup{u(μ)+F*(μ):μ∈M(X,T)}, Where P*(T, F) is the topological tail pressure of F, u(μ) is the tail entropy function of an entropy structure, F*(μ)=limn�'∞1/n.∫fndμ, and M(X,T) is the set of T-invariant measures on X. Moreover, the supremum can be achieved on the closure of the ergodic measures.At last, we extend the results above to the case of asymptotically sub-additive continuous potentials.