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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.16343 |
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| _version_ | 1866915206677921792 |
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| author | Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem |
| author_facet | Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem |
| contents | 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 $C_w$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16343 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lyapunov exponent attached to modular functions Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem Number Theory Dynamical Systems 11F03, 11J06 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 $C_w$. |
| title | A Lyapunov exponent attached to modular functions |
| topic | Number Theory Dynamical Systems 11F03, 11J06 |
| url | https://arxiv.org/abs/2503.16343 |