Estimations Of Hausdorff Dimension For C~1 Random Expanding Map

In this paper,we consider random dynamical system. Let (Ω. F,p,θ) be complete proba-bility space and M be compact smooth Riemannian manifold with dimension m.First, we give upper and lower bounds of Hausdorff dimension of invariant set A (denoted dim_H(?)) for C~1 expanding map f : M→M. We obtain the upper bound is the root of equationThen we use the formula of Brin-Katok local entropy to obtain the lower bound of Hausdorff dimension of ergodic invariant measure. Using this result, we obtain the lower bound of is the root of equationFor the invariance of A.we consider the iterated system and getFinally, we generalize these results to the pertubation of C~1 random expanding map f:Ω×M→Ω×M and obtain the similar results.