On fast Lyapunov spectra for Markov-R\'enyi maps
Abstract
In this paper, we study the multifractal analysis for Markov-R\'enyi 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\'enyi 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 [Theorem 1.1]FLWW13, [Theorem 1.2]LR, and [Theorem 1.2]FSW from the Gauss map to arbitrary Markov-R\'enyi 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\'enyi maps.
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