Event Type:

Number Theory Seminar

Date/Time:

Tuesday, January 17, 2017 - 16:00 to 17:00

Location:

BEXL 417

Local Speaker:

Abstract:

In 1981, H. Nakada defined his alpha-continued fractions, $T_{\alpha}: [(\alpha-1), \alpha] \to [(\alpha-1), \alpha)$ for $\alpha\in [0,1]$ such that $\alpha = 1$ is the Gauss (continued fractions) map, $x \mapsto 1/x - \lfloor 1/x \rfloor$. This family has been studied by numerous authors; with Arnoux, we showed that each $T_{\alpha}$ is a factor of the Poincare return map to a cross-section for the geodesic flow on $T^1(\text{PSL}_2(\mathbb Z)$, the unit tangent bundle of the modular surface.

With Calta and Kraaikamp, we define and study $\alpha$-type maps $T_{n,\alpha}$ for each of a countable number of (Fuchsian triangle) groups (thus playing the role of $\text{PSL}_2(\mathbb Z)$); these groups are defined over number fields (whose degree is unbounded). We show that the key property of orbit synchronization of the endpoints of the interval of definition holds for each $n$ on a set of $\alpha$ of full measure, and identify cross-sections for these $T_{n,\alpha}$.

In this first talk, we aim to motivate and make the results understandable.