The Lyapunov spectrum as the Newton-Raphson method for countable Markov interval maps

TL;DR

This paper extends the Lyapunov spectrum analysis to countable Markov interval maps with potential parabolic fixed points (MRL maps) by embedding the problem in a generalized thermodynamic formalism. It develops slide-function tools and a generalized Newton-Raphson mechanism (the S-Newton-Raphson map) to handle non-smooth pressure behavior and boundary issues, then connects the Lyapunov spectrum to the Legendre transform of the topological pressure $P(t)$ for the potential $-t\log|T'|$. A central result shows that, for $\alpha$ in the domain of $L$, the spectrum satisfies $L(\alpha)=\frac{1}{\alpha}\inf_{t\in\mathbb{R}}\{P(t)+t\alpha\}$, with the domain unbounded and $L(0)$ recovered as a limit when appropriate. The analysis blends approximation by finite subsystems, infimal convergence of pressures, and dimension-theoretic arguments to capture phase transitions and non-compact dynamics, thereby generalizing prior finite-branch results to the MRL setting and highlighting the role of parabolic phenomena in spectral regularity.

Abstract

We consider MRL maps (Markov-Renyi-Lüroth), a class of interval maps with infinitely many branches that can have parabolic fixed points. We prove that for every MRL map $T$, the Lyapunov spectrum can be expressed in terms of the Legendre transform of the topological pressure of $-t\log|T'|$, generalizing previous results in the area. We also show that the Lyapunov spectrum coincides with a function directly related to the Newton-Raphson method applied to the topological pressure of $-t\log|T'|$.

Figures (2)

• Figure 1: Types of slide functions.
• Figure 2: The Lyapunov spectrum graph of $MRL$ maps is depicted, where $a_P$ represents the limit as $t$ approaches $t_{\infty}^{+}$ of $P'(t)$ and $b_P$ represents the limit as $t$ approaches $\text{Dim}_{H}^{-}\Lambda$ of $P'(t)$. Additionally, for some $\alpha_{\text{min}} \geq 0$, we have $Dom(L) \subseteq [\alpha_{\text{min}},\infty)$. When $\alpha \geq -a_P$, the dashed segments correspond to $L(\alpha) = t_{\infty} + \frac{\lim_{t\rightarrow t_{\infty}^{+}}P(t)}{\alpha}$. The intersection of the horizontal dotted line with the $y$-axis represents the Hausdorff dimension of the set of points for which the Lyapunov exponent is not defined.

Theorems & Definitions (32)

• theorem 1.1
• definition 2.1
• definition 2.2
• proposition 2.3
• proof
• definition 2.4
• definition 2.5
• proposition 2.6
• proof
• corollary 2.7