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

TL;DR

This paper conducts a multifractal analysis of a Hénon-like map at the critical bifurcation point, linking Hausdorff dimension of level sets to entropy and Lyapunov exponents.

Contribution

It provides a formula for the Hausdorff dimension of Birkhoff average level sets in a non-uniformly hyperbolic Hénon-like map at bifurcation.

Findings

Hausdorff dimension formula for level sets

Decomposition of non-wandering points by Birkhoff averages

Analysis at the bifurcation where hyperbolicity breaks down

Abstract

We effect a multifractal analysis 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 decompose the set of non wandering points on the unstable manifold into level sets of Birkhoff averages of continuous functions, and derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures.