Outer Billiards, Digital Filters and Kicked Hamiltonians

TL;DR

This paper explores the relationship between outer billiards, digital filters, and kicked Hamiltonians, revealing conjugate dynamics and generalizations that enhance understanding of polygonal billiard systems and related maps.

Contribution

It introduces a digital filter map that exhibits conjugate dynamics to the outer billiards map for regular polygons and generalizes the Tangent map to 'step-k' versions, expanding analytical tools.

Findings

Digital filter map (Df) conjugate to Tangent map for even N-gons

Evidence of conjugacy between N and 2N-gons for odd N

Generalization of Tangent map to 'step-k' versions

Abstract

In 1978 Jurgen Moser suggested the outer billiards map (Tangent map) as a discontinuous model of Hamiltonian dynamics. A decade earlier, J.B. Jackson and his colleagues at Bell Labs were trying to understand the source of self-sustaining oscillations in digital filters. Some of the discrete mappings used to describe these filters show a remarkable ability to 'shadow' the Tangent map when the polygon in question is regular. In this paper we describe a specific digital filter map (Df) that appears to have dynamics which are conjugate to the Tangent map for a regular N-gon with N even. When N is odd, there is evidence of another conjugacy between the Tangent map dynamics of N and the matching 2N-gon, so a case like N = 7 can be studied with the Df map in the context of N = 14. This provides a many-fold increase in efficiency, and also allows us to generalize the Tangent map to obtain 'step-k' versions - which have dynamics that are unexplored. We also present some related maps, including Chua and Lin's 3-dimensional version of Df, an Analog to Digital Converter from Orla Feely, a sawtooth version of the Standard Map by Peter Ashwin and various kicked harmonic oscillators. All of these seem to shadow the Tangent map in some form. Mathematica code is provided for all mappings both here and at DynamicsOfPolygons.org.