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The Lyapunov spectrum for Schneider map on $p\mathbb{Z}_p$

Matias Alvarado, Nicolás Arévalo-Hurtado

TL;DR

The paper extends multifractal analysis to a non-Archimedean setting by studying the Lyapunov spectrum for the Schneider map on $p\mathbb{Z}_p$. It defines a geometric potential $\psi$ and the Lyapunov exponent $\lambda_p$, proving that the Lyapunov spectrum $L_p$ is real analytic on $[\log p,\infty)$ and giving an explicit closed form. By first analyzing compact truncations with bounded digits, the authors derive truncated spectra $L_{p,n}$ and then pass to the full system via infimal convergence of pressures, obtaining a complete description of $L_p(\alpha)$ and its domain. They further connect $\lambda_p(x)$ to the rate of convergence of Schneider continued fraction approximations, deriving exact dimension formulas for level sets of digits and approximation rates, thus providing a $p$-adic analogue of classical Gauss-map results and precise dimension formulas for natural level sets.

Abstract

We study the thermodynamic formalism associated with the Schneider map on the p-adic integers $p\mathbb{Z}_p$ . By introducing a geometric potential that captures the expansion of cylinder sets generated by the map, we define a Lyapunov exponent adapted to this non-Archimedean setting. We investigate the corresponding Lyapunov spectrum and show that it is real analytic on its natural domain. Moreover, we obtain an explicit closed formula for the spectrum. As a consequence, we recover and refine known results on the Hausdorff dimension of sets defined by a prescribed asymptotic arithmetic mean of the continued fraction digits. Finally, we relate the Lyapunov exponent to the exponential rate of convergence of rational approximations arising from truncations of the Schneider continued fraction expansion. This provides a $p$-adic analogue of classical results from Diophantine approximation and yielding precise dimension formulas for the associated level sets.

The Lyapunov spectrum for Schneider map on $p\mathbb{Z}_p$

TL;DR

The paper extends multifractal analysis to a non-Archimedean setting by studying the Lyapunov spectrum for the Schneider map on . It defines a geometric potential and the Lyapunov exponent , proving that the Lyapunov spectrum is real analytic on and giving an explicit closed form. By first analyzing compact truncations with bounded digits, the authors derive truncated spectra and then pass to the full system via infimal convergence of pressures, obtaining a complete description of and its domain. They further connect to the rate of convergence of Schneider continued fraction approximations, deriving exact dimension formulas for level sets of digits and approximation rates, thus providing a -adic analogue of classical Gauss-map results and precise dimension formulas for natural level sets.

Abstract

We study the thermodynamic formalism associated with the Schneider map on the p-adic integers . By introducing a geometric potential that captures the expansion of cylinder sets generated by the map, we define a Lyapunov exponent adapted to this non-Archimedean setting. We investigate the corresponding Lyapunov spectrum and show that it is real analytic on its natural domain. Moreover, we obtain an explicit closed formula for the spectrum. As a consequence, we recover and refine known results on the Hausdorff dimension of sets defined by a prescribed asymptotic arithmetic mean of the continued fraction digits. Finally, we relate the Lyapunov exponent to the exponential rate of convergence of rational approximations arising from truncations of the Schneider continued fraction expansion. This provides a -adic analogue of classical results from Diophantine approximation and yielding precise dimension formulas for the associated level sets.
Paper Structure (8 sections, 14 theorems, 169 equations)

This paper contains 8 sections, 14 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Restricting the Schneider map to bounded valuations
  4. Proof of main theorems
  5. Rational approximation for continued fraction expansions
  6. Proof of technical results.
  7. Proof of Lemma \ref{['BowenFormula']}
  8. Proof of Theorem \ref{['PrincipalPesinWeiss']}

Key Result

Theorem 1.1

The Lyapunov spectrum $L_{p}$ is real analytic on $[\log(p),\infty)$. For each $\alpha\geq \log p$ where $P(-t\log\psi)$ is the topological pressure of $-t\log\psi$ with respect to $T_p$. The infimum is attained at a unique $t_{\alpha}>0$ such that $\frac{d}{dt}P(-t\log \psi)|_{t=t_{\alpha}}=-\alpha$. Moreover, there exists a unique equilibrium state $\mu_{t_{\alpha}}$ for $-t_{\alpha}\log \psi$

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Example 1
  • ...and 17 more