Hyperbolic Sets: Past and Future

Abstract

We give a complete description of the dimension spectra of Birkhoff averages in a hyperbolic set of a conformal diffeomorphism, considering simultaneously Birkhoff averages into the past and into the future, i.e., both for negative and positive time. We emphasize that the description of these spectra is not a consequence of the results in Chapter 6. The main difficulty is that although the local product structure provided by the intersection of local stable and unstable manifolds is a Lipschitz homeomorphism with Lipschitz inverse, the level sets of Birkhoff averages are never compact. This causes their box dimension to be strictly larger than their Hausdorff dimension, and thus a product of level sets may have a Hausdorff dimension that a priori need not be the sum of the dimensions of the level sets. Instead, we construct explicitly nonivariant measures concentrated on each product of level sets having the appropriate pointwise dimension.