Lyapunov spectrum for Hénon-like maps at the first bifurcation

2014/05/08 by Hiroki Takahasi, Takahasi, Hiroki · 1 citation
Mathematics · Physics and Astronomy · #37D25 #37E30 #37G25 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:37D25 #msc:37E30 #msc:37G25

paper · pdf · doi:10.48550/arxiv.1405.1813

29 pages, 5 figures. Re-written for clarification

openalex publication_date 2014/05/08 · arxiv created 2015/01/31 · arxiv updated 2015/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a strongly dissipative Hénon-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.

Citations

• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Dimension, entropy and Lyapunov exponents

Cited by

• Birkhoff spectrum for Hénon-like maps at the first bifurcation

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