Lyapunov spectrum for H\'enon-like maps at the first bifurcation

TL;DR

This paper analyzes the Lyapunov spectrum of Hénon-like maps at the first bifurcation, providing a multifractal decomposition, a Hausdorff dimension formula, and continuity results for the spectrum.

Contribution

It offers a novel multifractal analysis of Hénon-like maps at bifurcation, including a formula for Hausdorff dimension and spectrum continuity.

Findings

Derived a Hausdorff dimension formula for level sets.

Proved the continuity of the Lyapunov spectrum.

Showed the set with non-existent exponents has full Hausdorff dimension.

Abstract

For a strongly dissipative H\'enon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries a full Hausdorff dimension.