Scaling Properties Of Hyperbolic Measures

. In this article we consider a class of maps which includes C 1+ff diffeomorphisms as well as invertible and nonivertible maps with piecewise smooth singularities. We prove a general scaling result for any hyperbolic measure which is invariant for a map from our class. The existence of the pointwise dimension and the Brin-Katok local entropy formula are special cases of our scaling result. 1. Introduction Let dm (x) := lim r!0 log m(B(x; r)) log r (1) whenever the limit exists. For ergodic measures, if the limit exists, then it is almost everywhere constant and is called the pointwise dimension of the measure m: Recently Barreira, Pesin and Schmeling proved that if f is a C 1+ff diffeomorphism of a compact Riemannian manifold then the pointwise dimension exists almost everywhere for any f-invariant hyperbolic measure m [1]. The proof of this result uses the local geometric structure of hyperbolic measures (local product structure). Let f : (X; d) ! (X; d) a continuous mappin..