A Geometric Dynamical System with Relation to Billiards
Vol. 32 (2025), No. 2, Page 127–156.
Everett, Samuel
A Geometric Dynamical System with Relation to Billiards
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Abstract:
We introduce a geometric dynamical system where iteration is defined as a cycling composition of a finite collection of geometric maps, which act on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated function systems, as well as billiards with modified reflection laws. We provide conditions under which this dynamical system generates periodic orbits, and use this result to prove the existence of closed nonsmooth curves over $\mathbb{R}^2$ which satisfy particular structural constraints with respect to a space of intersecting lines in the plane.
Keywords: Piecewise continuous map, contraction mapping, periodic orbits, billiards.
Mathematics Subject Classification (2020): 37E99, 37C83, 51N20.
Received: 2022-01-30
