2025/03/20 by Paloma Bengoechea, Sebastián Herrero, Bengoechea, Paloma +3 · 2 citations
Computer Science · #11F03 #11J06 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Control Systems and Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2503.16343
openalex publication_date 2025/03/20 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
To each weakly holomorphic modular function f\not ≡ 0 for SL(2,ℤ), which is non-negative on the geodesic arc \eit : π/3≤ t≤ 2π/3\, we attach a GL(2,ℤ)-invariant map Λf:ℙ1(ℝ)→ ℝ that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between 0 and Λf((1+√(5))/(2)) 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 Λf(w) equals the real part of the cycle integral of f along the associated geodesic Cw on the modular surface, normalized with the word length of the associated hyperbolic matrix Aw as a word in the generators T=(\beginsmallmatrix 1 & 1 0 & 1 \endsmallmatrix) and V=(\beginsmallmatrix 1 & 0 1 & 1 \endsmallmatrix). 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 Cw.