On fast Lyapunov spectra for Markov-R\'{e}nyi maps

TL;DR

This paper analyzes the fast Lyapunov spectrum of Markov-Rényi maps, revealing it as a piecewise constant function with potential discontinuities, extending previous results from Gauss maps to a broader class of interval maps.

Contribution

It introduces a geometric approach to study the fast Lyapunov spectrum without distortion assumptions, generalizing prior work to Markov-Rényi maps with complex features.

Findings

Fast Lyapunov spectrum is piecewise constant.

Spectrum may have discontinuity at infinity.

Extended results from Gauss maps to Markov-Rényi maps.

Abstract

In this paper, we study the multifractal analysis for Markov-R\'{e}nyi 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-R\'{e}nyi 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-R\'{e}nyi 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-R\'{e}nyi maps.