Ergodic Optimization Of Asymptotically Additive Potentials

Let T:X?X be a continuous map on a compact metric space X.A sequence of potentials ?={?n}n?1 is asymptotically additive if for any ?>0,there exists a continuous function g? satisfying(?) where ||f||=(?)f(x)|.This paper considers some problems about ergodic optimiza-tion in the quotient vector space S,which consists of the equivalence classes of these asymptotically additive potentials.More specifically,this paper firstly proves that,for a topological vector space which is densely and continuously embedded in S,the set composed by all elements in this space which have a unique maximizing measure is a residual set of this space.Secondly,if T is transitive,and hyperbolic with local prod-uct structure,then the set composed by all elements in S for which every maximizing measure has full support is a residual set of S.Thirdly,if T is an expanding map,then the set composed by all elements in S whose unique maximizing measure has zero metric entropy is a residual set of S.In the end,if(X,T)satisfies ASP and MCGBP properties,and T is Lipschitz continuous,then the set composed by all elements in S whose unique maximizing measure is supported on a periodic orbit is a dense set of S.