Pointwise Dimension For Regular Hyperbolic Measures For Endomorphisms

this article we consider hyperbolic measures which are invariant for endomorphisms and show the corresponding result: the pointwise dimension exists almost everywhere. The two dimensional version of our theorem was proven in our earlier article [17], the result of this article hold in any finite dimension. The existence of the pointwise dimension has several important implications, for endomorphisms as well as for diffeomorphisms. The Hausdorff dimension and the box dimension of the measure as well as some other characteristics of dimension type of the measure coincide. We refer the reader to the reference [11] for a survey of these characteristics. To prove our theorem we use a solenoidal construction to get an invertible map with singularities. This map will be in the class originally studied by Pesin [9]. We use the generalized theory of Ledrappier and Young [7] and the result of Barreira, Pesin and Schmeling [3]. The results of Ledrappier and Young give a relation between Lyapunov exponents, Hausdorff dimension and entropy of an invariant measure. The Ledrappier, Young theory is stated fo