摘要
Let f be a continuous transformation on a compact, finite-dimensional manifold M, and φ a continuous function on M. This paper establishes the following formula：ess sup lim sup n→∞1/nφn（x）=sup{∫φdμ｜μ∈Of}≤lim sup n→∞1/n ess supφn（x）,where ess sup denotes the essential supremum taken against the Lebesgue measure,φn（x）=∑i=0^n-1φ（f^ix）and Of is the set of observable measures. Examples are provided to illustrate that the inequality could be an equality or strict. Moreover, if μ is the unique maximizing observable measure for φ, it is weakly statistical stable.
Let f be a continuous transformation on a compact, finite-dimensional manifold M, and φ a continuous function on M. This paper establishes the following formula：ess sup lim sup n→∞1/nφn（x）=sup{∫φdμ｜μ∈Of}≤lim sup n→∞1/n ess supφn（x）,where ess sup denotes the essential supremum taken against the Lebesgue measure,φn（x）=∑i=0^n-1φ（f^ix）and Of is the set of observable measures. Examples are provided to illustrate that the inequality could be an equality or strict. Moreover, if μ is the unique maximizing observable measure for φ, it is weakly statistical stable.
基金
Supported by NSFC(Grant No.11371271)
the Priority Academic Program Development of Jiangsu Higher Education Institutions
