The paper studies the dynamical properties of the outer billiard map on regular polygons. The main result is that outer billiard on any regular N-gon with N > 4 and N ≠ 6 has aperiodic points, answering a question posed by R. Schwartz.
The key insights are:
Outer billiard maps on regular polygons can be viewed as piecewise rotations with unbounded domains. To study their dynamics, the authors focus on a bounded invariant region called the "vassal polygon".
On the vassal polygon, the second iterate of the outer billiard map becomes a polygon exchange transformation, which allows the use of tools from the theory of dynamical valuations and scissors congruence invariants.
For N divisible by 4, the authors construct a non-vanishing dynamical Hadwiger invariant, which implies the existence of aperiodic points.
For N giving remainder 2 when divided by 4, they reduce the problem to studying an associated interval exchange transformation and use the classical Sah-Arnoux-Fathi invariant to rule out periodicity.
The paper also discusses previous results on outer billiards, the connections to piecewise isometries and valuations, and the methods used to tackle the problem.
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by Anton Belyi,... a las arxiv.org 10-02-2024
https://arxiv.org/pdf/2311.09643.pdfConsultas más profundas