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日時: 2025年5月30日(金) 15:30-16:30
Date: May 30, 2025 15:30-16:30
会場:東京大学大学院数理科学研究科 大講義室
Place: auditorium, Graduate School of Math. Sci. Bldg.
John A G Roberts 氏（UNSW Sydney / 東京大学大学院数理科学研究科）
John A G Roberts (School of Mathematics and Statistics, UNSW Sydney / Graduate School of Mathematical Sciences, The University of Tokyo)
Arithmetic and geometric aspects of the (symbolic) dynamics of piecewise-linear maps (ENGLISH)
We study a family of planar area-preserving maps, described by different SL(2,R) matrices on the right and left half-planes. Such maps, studied extensively by Lagarias and Rains in 2005, can support periodic and quasiperiodic dynamics with a foliation of the plane by invariant curves. The parameter space is two dimensional (the parameters being the traces of the two matrices) and the set of parameters for which an initial condition on the half-plane boundary returns to it are algebraic “critical” curves, described by the symbolic dynamics of the itinerary between the boundaries. An important component of the planar dynamics is the rotational dynamics it induces on the unit circle. The study of the arithmetic, algebraic, and geometric aspects of the planar and circle (symbolic) dynamics has connections to various parts of number theory and geometry, which I will mention. These include: Farey sequences; continued fraction expansions and continuant polynomials; the character variety of group representations in SL(2,C) and PSL(2,C); and the group of polynomial diffeomorphisms of C3 preserving the Fricke-Vogt invariant x2 + y2 + z2 - xyz.
This is joint work with Asaki Saito (Hakodate) and Franco Vivaldi (London)..