Gibbs Measure Of Cookie-cutter Set

Let X(?)C R be a bounded non-empty closed interval,and let X1 and X2 be disjoint subintervals of X.Let f:X1 ? X2 ? X be such that X1 and X2 are each mapped bijectively onto X.And f has a continuous derivative and is expanding.The set E = {x ? X|fk(x)is defined and in X1 U X2 for all k = 0,1,2,…},where fk is the k-th iterate of f,is called a cookie-cutter set by f.This paper based on the known results gives a method of constructing Gibbs measure supported by E,which is different from Falconer's method in the context of[10].Firstly,the Ruelle-Perron-Proebenius theorem(it has two conclusions)is proved by using the transfer operator L? induced by the Lipschits function ?.From the con-clusion(?)the transfer operator L? and its conjugate operator have characteristic function,h and v respectively.Secondly,let ? hv.From the conclusion(?)of Ruelle-Perron-Probenius theo-rem,we know ? is f-mixing.And it's easy to proof ? is f-invariant.At last,using the two properties of ?,we prove that ? is the Gibbs measure supported by E and Gibbs measure is unique.