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A Lyapunov exponent attached to modular functions

Paloma Bengoechea, Sebastián Herrero, Özlem Imamoglu

TL;DR

This work constructs a GL(2,Z)–invariant Lyapunov-type exponent Λ_f attached to any nonzero weakly holomorphic modular function f for SL(2,Z). It shows Λ_f vanishes on rational arguments, is invariant under SL(2,Z) action, and attains values in [0, Λ_f(φ)], with φ the golden ratio; for quadratic irrationals, Λ_f(x) equals the normalized real part of a cycle integral along the corresponding geodesic. The paper establishes a precise link between Λ_f and cycle integrals via explicit kernel representations, and uses Farey and Markov trees to analyze the variation of Λ_f on Markov irrationals, proving that tilde Λ_f is continuous, decreasing, and convex on [0,1/2]. It yields a lower bound Λ_f(𝔐)≥Λ_f(ψ) for Markov irrationals and, more generally, shows the range on Markov numbers lies between the silver and golden ratios; the results connect to Kaneko’s val function and stabilize via almost invariance/additivity properties of cycle integrals. Methodologically, the paper blends cycle-integral analysis, continued fraction dynamics, and stable-norm ideas to obtain a detailed topological and convex-analytic picture of Λ_f on the real line and on Markov trees.

Abstract

To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : π/3\leq t\leq 2π/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map $Λ_f:\mathbb{P}^1(\mathbb{R})\to \mathbb{R}$ that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between $0$ and $Λ_f\left(\frac{1+\sqrt{5}}{2}\right)$ and it gives an increasing convex function on the Markov irrationalities when ordered using their parametrization by Farey fractions in $[0,1/2]$. In the case of quadratic irrationals $w$ with purely periodic continued fraction expansion, the value $Λ_f(w)$ equals the real part of the cycle integral of $f$ along the associated geodesic $C_w$ on the modular surface, normalized with the word length of the associated hyperbolic matrix $A_w$ as a word in the generators $T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\right)$. These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular $j$ function when normalized by the hyperbolic length of the geodesic $C_w$.

A Lyapunov exponent attached to modular functions

TL;DR

This work constructs a GL(2,Z)–invariant Lyapunov-type exponent Λ_f attached to any nonzero weakly holomorphic modular function f for SL(2,Z). It shows Λ_f vanishes on rational arguments, is invariant under SL(2,Z) action, and attains values in [0, Λ_f(φ)], with φ the golden ratio; for quadratic irrationals, Λ_f(x) equals the normalized real part of a cycle integral along the corresponding geodesic. The paper establishes a precise link between Λ_f and cycle integrals via explicit kernel representations, and uses Farey and Markov trees to analyze the variation of Λ_f on Markov irrationals, proving that tilde Λ_f is continuous, decreasing, and convex on [0,1/2]. It yields a lower bound Λ_f(𝔐)≥Λ_f(ψ) for Markov irrationals and, more generally, shows the range on Markov numbers lies between the silver and golden ratios; the results connect to Kaneko’s val function and stabilize via almost invariance/additivity properties of cycle integrals. Methodologically, the paper blends cycle-integral analysis, continued fraction dynamics, and stable-norm ideas to obtain a detailed topological and convex-analytic picture of Λ_f on the real line and on Markov trees.

Abstract

To each weakly holomorphic modular function for , which is non-negative on the geodesic arc , we attach a -invariant map that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between and and it gives an increasing convex function on the Markov irrationalities when ordered using their parametrization by Farey fractions in . In the case of quadratic irrationals with purely periodic continued fraction expansion, the value equals the real part of the cycle integral of along the associated geodesic on the modular surface, normalized with the word length of the associated hyperbolic matrix as a word in the generators and . These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular function when normalized by the hyperbolic length of the geodesic .

Paper Structure

This paper contains 21 sections, 36 theorems, 192 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Cycle integrals
  4. Elementary properties
  5. Cycle of a purely periodic quadratic irrational
  6. Almost invariance
  7. Almost additivity
  8. A formula for $\mathrm{Re}(I_f(A))$
  9. An upper bound for $\mathrm{Re}(I_f(A))$
  10. First properties of $\Lambda_f$
  11. Proof of Theorem \ref{['thm:first_properties_Lambda_f']}
  12. Proof of Theorem \ref{['thm:Lambda_f_on_quad_irrationals']}
  13. Values attained by $\Lambda_f$
  14. Proof of Theorem \ref{['thm:values_of_Lambda_f']}
  15. Minimum over Markov irrationalities
  16. ...and 6 more sections

Key Result

Theorem 1.4

The limit superior in eq:def_lambda_f_intro is finite for all $x\in (0,\infty)$ and $\Lambda_f$ extends to a function $\Lambda_f:\mathbb{P}^1(\mathbb{R})\to \mathbb{R}$ satisfying Moreover, $\Lambda_f(x)=0$ for all $x\in \mathbb{P}^1(\mathbb{Q})$.

Figures (7)

  • Figure 1: Farey tree $\mathcal{F}_{\frac{0}{1},\frac{1}{0}}$.
  • Figure 2: Image of $\{0,1,\infty\}$ under the matrix $A=\left(cadb\right).$
  • Figure 3: Labeled Farey tree.
  • Figure 4: The Markov tree.
  • Figure 5: Values of $\tilde{\Lambda}_j$ on the first 257 points in $\mathcal{F}_{\frac{0}{1},\frac{1}{2}}$ on the left plot, and on the first 1025 points on the right plot.
  • ...and 2 more figures

Theorems & Definitions (75)

  • Example 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 65 more