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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.16291 |
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| _version_ | 1866912441704644608 |
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| author | Fang, Lulu Moreira, Carlos Gustavo Wang, Zhichao Zhang, Yiwei |
| author_facet | Fang, Lulu Moreira, Carlos Gustavo Wang, Zhichao Zhang, Yiwei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16291 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On fast Lyapunov spectra for Markov-Rényi maps Fang, Lulu Moreira, Carlos Gustavo Wang, Zhichao Zhang, Yiwei Dynamical Systems 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. |
| title | On fast Lyapunov spectra for Markov-Rényi maps |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.16291 |