A Variational Principle Of Entropy On Subsets For Fixed-points Free Flows

In this paper,we study a variational principle between Bowen's topological entropy and measure-theoretic entropy on subsets for compact metric flows without fixed points.Firstly,we make a review of the classic definitions of measure-theoretic entropy,topological entropy and topological pressure for discrete topological dynamical sys-tems,as well as the basic concepts of Hausdorff dimension and packing dimension in fractal geometry theory.Secondly,we introduce the concept of Bowen's topological entropy,packing topological entropy,lower and upper entropy on subsets,as well as a result of the variational principle of Bowen's topological entropy and lower measure-theoretic entropy,packing topological entropy and upper measure-theoretic entropy on subsets by D.J.Feng and W.Huang.Then,we prove a covering lemma in the sense of reparameterization on flows,by using the reparameterization technique to characterize some basic properties of repa-rameterization balls on flows,which is a variation of the classical 5r-covering lemma in fractal geometry theory.Finally,following Feng and Huang's technical line,we s-tudy the relationship between Bowen's topological entropy and local measure-theoretic entropy on subsets for compact metric flows without fixed points,and we obtain the following variational principle:Let(X,d)be a compact metric flow without fixed points.If K is a non-empty compact subset of X,then htopB(?,K)=sup{h?(?):??M(X),?(K)=1}.