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談話会・数理科学講演会
過去の記録 ～05/18｜次回の予定｜今後の予定 05/19～
| 担当者 | 会田茂樹，大島芳樹，志甫淳（委員長），高田了 |
|---|---|
| セミナーURL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium/index.html |
2025年05月30日(金)
15:30-16:30 数理科学研究科棟(駒場) 大講義室(auditorium)号室
John A G Roberts 氏 (UNSW Sydney / 東京大学大学院数理科学研究科)
Arithmetic and geometric aspects of the (symbolic) dynamics of piecewise-linear maps (English)
John A G Roberts 氏 (UNSW Sydney / 東京大学大学院数理科学研究科)
Arithmetic and geometric aspects of the (symbolic) dynamics of piecewise-linear maps (English)
[ 講演概要 ]
We study a family of planar area-preserving maps, described by different $SL(2,\mathbb{R})$ 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 $SL(2, \mathbb{C})$ and $PSL(2, \mathbb{C})$; and the group of polynomial diffeomorphisms of $\mathbb{C}^3$ preserving the Fricke-Vogt invariant $x^2 + y^2 + z^2 - xyz$.
This is joint work with Asaki Saito (Hakodate) and Franco Vivaldi (London).
We study a family of planar area-preserving maps, described by different $SL(2,\mathbb{R})$ 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 $SL(2, \mathbb{C})$ and $PSL(2, \mathbb{C})$; and the group of polynomial diffeomorphisms of $\mathbb{C}^3$ preserving the Fricke-Vogt invariant $x^2 + y^2 + z^2 - xyz$.
This is joint work with Asaki Saito (Hakodate) and Franco Vivaldi (London).
