Event Type:

Number Theory Seminar

Date/Time:

Tuesday, November 6, 2018 - 16:00 to 17:00

Location:

Bexell 102

Local Speaker:

Abstract:

Regular continued fractions appear in many settings. Gauss (1812) determined an invariant probability measure for their underlying interval map, but never explained how. Ito-Nakada-Tanaka (1977) gave a simple method, using a 2-dimensional system. Nakada (1981) introduced an infinite family of interval maps T_{alpha}, called alpha-continued fractions. By 2012, one could use 2-dimensional systems to find invariant measures and more for all T_{alpha}. Arnoux-S (2013) showed that each T_{alpha} is induced by the first return map under the geodesic flow to a cross-section of the (three dimensional) unit tangent bundle of the surface uniformized by the modular group. I'll explain some of these matters and report on work with Calta and Kraaikamp on alpha-like families of maps for other groups.