Hausdorff Dimension And Entropy Of The Exceptional Sets Of Average Conformal Maps

We consider the Hausdorff dimension and entropy of the exceptional sets of C1+αaverage conformal maps,both non-invertible and invertible case,and their relation with the Hausdorff dimension and entropy of the system.Precisely,for the non-invertible case,we proved that if the entropy on a set A is less than that of the system,then we have the entropy on the exceptional set of A equals that of the system.Given a hyperbolic ergodic measure μ,if the Hausdorff dimension of the set A is less than that of the measure μ,then we have the entropy on the exceptional set of A is greater than or equal to the measure theoretical entropy.Further more,if the topological entropy on the set A is less than the measure theoretical entropy,we get that the Hausdorff dimension of the exceptional set of A is greater than or equal to that of measure μ.When the Hausdorff dimension of the set A is less than the dynamical dimension,we have the Hausdorff dimension of the exceptional set of the set A is greater than or equal to the dynamical dimension.For the invertible case,we get similar results.