Fractal Weyl bounds and Hecke triangle groups

Eigenfunctions of transfer operators and automorphic forms for Hecke triangle groups of infinite covolume

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.

Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, M. Möller and the author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $Z(s) = \det(1-\mc L_s^+)\det(1-\mc L_s^-)$. In this article we show that the operator families $\mc L_s^\pm$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $s\in\C$, $\Rea s=\tfrac12$, the operator $\mc L_s^+$ (resp. $\mc L_s^-$) has a 1-eigenfunction if and on…

Hecke triangle groups, transfer operators and Hausdorff dimension

We consider the family of Hecke triangle groups $ Γ_{w} = \langle S, T_w\rangle $ generated by the Möbius transformations $ S : z\mapsto -1/z $ and $ T_{w} : z \mapsto z+w $ with $ w > 2.$ In this case the corresponding hyperbolic quotient $ Γ_{w}\backslash\mathbb{H}^2 $ is an infinite-area orbifold. Moreover, the limit set of $ Γ_w $ is a Cantor-like fractal whose Hausdorff dimension we denote by $ δ(w). $ The first result of this paper asserts that the twisted Selberg zeta function $ Z_{Γ_{ w}}(s, ρ) $, where $ ρ: Γ_{w} \to \mathrm{U}(V) $ is an arbitrary finite-dimensional unitary represent…

通过转移算子研究 Hecke 三角群的谱与动力学不变量

Spectral and dynamical invariants of Hecke triangle groups via transfer operators

本文利用转移算子将 Hecke 三角群的 Selberg zeta 函数近似为有限维矩阵行列式，并据此高精度计算 Hausdorff 维数、Ruelle zeta 函数零点值及平凡零点阶数估计。

Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations

Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which Quantum Unique Ergodicity holds.