Extreme value theorem for geodesic flow on the quotient of the theta group

Abstract

We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$.

Figures (8)

• Figure 1: A fundamental domain $\mathcal{F}$ of $\Theta$ and the quotient surface $\mathcal{M}=\Theta\backslash \mathbb{H}^2$.
• Figure 2: The graph of $T$.
• Figure 3: The graph of the dual SCF map $\widehat{T}$
• Figure 4: Tessellation $\mathcal{T}$ with the quadrilateral $\mathcal{Q}$ whose vertices are $-1,0,1,\infty$.
• Figure 5: Illustration of the case $s=e$: the geodesic $\gamma$ successively crosses the $k-1$ vertical lines $\ell(2i+1,\infty)$, and meets $C_{(k,\varepsilon)_s}$ at time $t_k$, which is the first return time.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: spliced continued fraction expansion
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Definition 2.5