Upper estimates for stable dimensions of fractal sets with variable number of foldings

TL;DR

This paper derives upper estimates for the stable Hausdorff dimension of saddle-type fractals with variable preimage counts using thermodynamic formalism, revealing conditions for constant stable dimension and behavior similar to homeomorphisms.

Contribution

It introduces a method to estimate stable dimensions of hyperbolic fractals with variable preimages, extending prior results to more complex, non-constant preimage scenarios.

Findings

Stable dimension estimates depend on a lower bound of the preimage counting function.

Maximal stable dimension implies constancy of the dimension and preimage count on a dense subset.

If stable dimension equals the stable similarity dimension, the map behaves like a homeomorphism.

Abstract

For a hyperbolic map f on a saddle type fractal Lambda with self-intersections, the number of f- preimages of a point x in Lambda may depend on x. This makes estimates of the stable dimensions more difficult than for diffeomorphisms or for maps which are constant-to-one. We employ the thermodynamic formalism in order to derive estimates for the stable Hausdorff dimension function delta^s on Lambda, in the case when f is conformal on local stable manifolds. These estimates are in terms of a continuous function on Lambda which bounds the preimage counting function from below. As a corollary we obtain that if delta^s attains its maximal possible value in Lambda, then the stable dimension is constant throughout Lambda, whereas the preimage counting function is constant on at least an open and dense subset of Lambda. In particular, this shows that if at some point in Lambda, the stable dimension is equal to the analogue of the similarity dimension in the stable direction at that point, then f behaves very much like a homeomorphism on Lambda. Finally we also obtain results about the stable upper box dimension for these type of fractals. We end the paper with a discussion of two explicit examples.