A geometric dynamical system with relation to billiards

TL;DR

This paper introduces a new geometric dynamical system based on cycling compositions of maps on lines in the plane, inspired by billiards and iterated function systems, and explores conditions for periodic orbits and structural properties.

Contribution

It presents a novel dynamical system on lines in the plane, linking billiard dynamics with geometric iteration and establishing conditions for periodic orbits and closed curves.

Findings

Conditions for periodic orbits are established.

Existence of closed nonsmooth curves satisfying structural constraints.

The system connects billiard dynamics with geometric iteration.

Abstract

We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting 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.