Multifractal Structure of Two-dimensional Horseshoes

We give a complete description of the dimension spectra of Birkhoff averages on a hyperbolic set of a surface diffeomorphism. The main novelty is that we are able to consider simultaneously Birkhoff averages into the future and into the past, i.e., both for positive and negative time. We emphasize that the description of these spectra is not a consequence of the available results in the case of Birkhoff averages simply into the future (or into the past). The main difficulty is that although the local product structure provided by the intersection of stable and unstable manifolds is bi-Lipschitz equivalent to a product, the level sets of the Birkhoff averages are never compact (this causes their box dimension to be strictly larger than their Hausdorff dimension), and thus the product of level sets may have a dimension that need not be the sum of the dimensions of the sets. Instead we construct explicitly noninvariant measures concentrated on each product of level sets with the appropriate pointwise dimension. We also consider the higher-dimensional case of more than one Birkhoff average, as well as the case of ratios of Birkhoff averages.