Abstract: In hyperbolic dynamics the regularity of the invariant foliations (holonomies) plays an important role in the study of dynamically defined fractals. In higher dimensional cases the lack of such regularity is a major complication in the understanding of simpler fractal invariants such as Hausdorff dimension. In this talk we focus on properties of the dynamics which implies this lack of regularity of invariant foliations and how to bypass these difficulties (generically) for higher dimensional horseshoes.