Stable sets of certain non-uniformly hyperbolic horseshoes have the expected dimension

TL;DR

This paper proves that the stable and unstable sets of certain non-uniformly hyperbolic horseshoes have the expected Hausdorff dimension, confirming a conjecture for specific surface diffeomorphisms undergoing heteroclinic bifurcations.

Contribution

It establishes the Hausdorff dimension of stable and unstable sets in non-uniformly hyperbolic horseshoes arising from heteroclinic bifurcations, confirming a prior conjecture.

Findings

Stable and unstable sets have the conjectured Hausdorff dimension.

Results apply to heteroclinic bifurcations with horseshoes of Hausdorff dimension less than 22/21.

Confirms the dimension conjecture for a class of surface diffeomorphisms.

Abstract

We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated to horseshoes with Hausdorff dimension $<22/21$ of conservative surface diffeomorphisms.