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arXiv · 2506.16291

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

Lulu Fang·Carlos Gustavo Moreira·Zhichao Wang·Yiwei Zhang
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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.

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Lulu Fang, Carlos Gustavo Moreira, Zhichao Wang, Yiwei Zhang. 2025-06-19. On fast Lyapunov spectra for Markov-R\'{e}nyi maps. https://arxiv.org/abs/2506.16291

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