On fast Lyapunov spectra for Markov-R\'{e}nyi 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.
Keywords
Cite
@article{arxiv.2506.16291,
title = {On fast Lyapunov spectra for Markov-R\'{e}nyi maps},
author = {Lulu Fang and Carlos Gustavo Moreira and Zhichao Wang and Yiwei Zhang},
journal= {arXiv preprint arXiv:2506.16291},
year = {2025}
}
Comments
41 pages 3 Figures