On fast Lyapunov spectra for Markov-Rényi maps

2025/06/19 by Fang, Lulu, Moreira, Carlos Gustavo, Wang, Zhichao +1
Mathematics · Neuroscience · Physics and Astronomy · #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Neural dynamics and brain function

paper · pdf · doi:10.48550/arxiv.2506.16291

openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the multifractal analysis for Markov-Rényi maps, which form a canonical class of piecewise differentiable interval maps, with countably many branches and may contain a parabolic fixed point simultaneously, and do not assume any distortion hypotheses. We develop a geometric approach, independent of thermodynamic formalism, to study the fast Lyapunov spectrum for Markov-Rényi maps. Our study can be regarded as a refinement of the Lyapunov spectrum at infinity. We demonstrate that the fast Lyapunov spectrum is a piecewise constant function, possibly exhibiting a discontinuity at infinity. Our results extend the works in \cite[Theorem 1.1]FLWW13, \cite[Theorem 1.2]LR, and \cite[Theorem 1.2]FSW from the Gauss map to arbitrary Markov-Rényi maps, and highlight several intrinsic differences between the fast Lyapunov spectrum and the classical Lyapunov spectrum. Moreover, we establish the upper and lower fast Lyapunov spectra for Markov-Rényi maps.

Citations

• Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches
• Upper and lower fast Khintchine spectra in continued fractions
• On the fast Khintchine spectrum in continued fractions
• Multifractal analysis for expanding interval maps with infinitely many branches
• On Khintchine exponents and Lyapunov exponents of continued fractions
• Multifractal analysis of non-uniformly hyperbolic systems
• The Lyapunov spectrum of some parabolic systems
• Geometry of limit sets for expansive Markov systems
• On the topological entropy of saturated sets
• On the variational principle for the topological entropy of certain non-compact sets
• Variational principles and mixed multifractal spectra
• Fractal Geometry: Mathematical Foundations and Applications.

Related

• The Lyapunov spectrum as the Newton-Raphson method for countable Markov interval maps
• Multifractal analysis of the Lyapunov exponent for random graph directed Markov systems
• Simplicity of Lyapunov spectra: a sufficient criterion
• Lyapunov spectrum for Hénon-like maps at the first bifurcation
• Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches