论文标题
Hecke Triangle组，转移操作员和Hausdorff Dimension
Hecke triangle groups, transfer operators and Hausdorff dimension
论文作者
论文摘要
我们考虑由Möbius变换$ s生成的Hecke Triangle组$γ_{w} = \ langle s，t_w \ rangle $ $ s：z \ mapsto -1/z $和$ t_ $ and $ t_ {w}：z \ mapsto z+w $ with $ w> 2. γ_{w} \ BackSlash \ Mathbb {H}^2 $是无限型区域的ORBIFOLD。此外，$γ_W$的极限集是一种类似cantor的分形，我们用$δ表示的hausdorff尺寸（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.该结果有许多应用程序。我们研究了零在半套$ \ mathrm {re}（s）> \ frac {1} {2} $ selberg zeta函数的零群函数$（γ_w^^n）_ {n \ in \ mathbb {n} $γ__的$γ___的$ stelbb {这些零对应于关联双曲线表面上的laplacian的特征值$ x_w^n =γ_w^n \ backslash \ mathbb {h}^2 $。我们表明，经典的Selberg zeta函数$ z_ {γ_W}（s）$可以通过有限矩阵的决定因素近似，其条目是根据Riemann Zeta函数明确给出的。此外，我们证明了Hausdorff Dimension $δ（W）$的渐近扩展为$ W \ to \ infty $。
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 $w\to \infty$.
