Unstable directions and dimension for a class of skew products with overlaps

TL;DR

This paper investigates skew products with overlaps, demonstrating that unstable manifolds depend on prehistories and analyzing the Hausdorff dimension of fibers using inverse pressure, Cantor set thickness, and preimage bounds.

Contribution

It introduces new results on the dependence of unstable manifolds on prehistories and provides dimension estimates for fibers in skew products with overlaps.

Findings

Unstable manifolds depend on prehistories even under perturbations.

Hausdorff dimension of fibers can be estimated using inverse pressure and Cantor set thickness.

Bounds for the preimage counting function are established.

Abstract

We study a class of skew products with overlaps in fibers and show that in this case the unstable manifolds really depend on prehistories, even for perturbations of the original maps. We also give several results about the Hausdorff dimension of the fibers of the respective locally maximal invariant set, by using the inverse pressure, the thickness of Cantor sets and some bounds for the preimage counting function.