Hecke三角群、迁移算子与豪斯多夫维数
Hecke triangle groups, transfer operators and Hausdorff dimension
摘要：
我们考虑由Möbius变换$ S : z\mapsto -1/z $和$ T_{w} : z \mapsto z+w $与$ w > 2.$生成的Hecke三角形群族$ Γ_{w} = \langle S, T_w\rangle $。在这种情况下，相应的双曲商$ Γ_{w}\backslash\mathbb{H}^2 $是一个无限区域的orbifold。本文的第一个结果证明了扭Selberg zeta函数$ Z_{Γ_{ w}}(s, ρ) $（$ ρ: Γ_{w} \to \mathrm{U}(V) $是任意有限维酉表示）可以实现为Mayer型转移算子的豪斯多夫维数行列式.这一结果有许多应用。研究了$Γ_w$的一类特殊子群$( Γ_w^n )_{n\in \mathbb{N}} $的Selberg zeta函数的半平面$\mathrm{Re}(s) > \frac{1}{2}$中零点的分布.这些零点对应于拉普拉斯算子在相关双曲曲面$X_w^n = Γ_w^n \backslash \mathbb{H}^2$上的本征值。我们表明，经典的Selberg zeta函数$Z_{Γ_w}(s)$可以近似的有限矩阵，其条目明确给出的黎曼zeta函数的行列式。此外，我们证明了一个渐近展开的豪斯多夫维数$δ(w)$为$w\to \infty$。
英文摘要：
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 representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane $\mathrm{Re}(s) > \frac{1}{2}$ of the Selberg zeta function of a special family of subgroups $( Γ_w^n )_{n\in \mathbb{N}} $ of $Γ_w$. These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $X_w^n = Γ_w^n \backslash \mathbb{H}^2$. We show that the classical Selberg zeta function $Z_{Γ_w}(s)$ can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension $δ(w)$ as $w\to \infty$.
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