Unstable directions and fractal dimension for skew products with overlaps in fibers

Abstract

A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set Λ (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of Λ for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in Λ with two preimages belonging to Λ, as well as uncountably many points having only one preimage in Λ. In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on Λ, as well as far from being constant-to-1 maps on Λ.