Abstract

Circles will be characterized by some properties of billiard ball trajectories. The theory of parallels and the parallel axiom play important roles in the geometry of the configuration space. Those characterizations are concerned with Bialy's theorem which is a partial answer to Birkhoff's conjecture.

Highlights

  • Let C be a smooth simple closed and strictly convex curve with length L in the Euclidean plane E and let c : R ! E be its representation by arclength, namely jc_ðtÞj 1⁄4 1 for any t A R where R is the set of all real numbers

  • Shinetsu Tamura and Nobuhiro Innami space W is foliated by closed curves invariant under the billiard ball map j

  • Let aðxÞ denote the slope of the billiard ball trajectory determined by x for x A W

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Summary

Introduction

Let C be a smooth simple closed and strictly convex curve with length L in the Euclidean plane E and let c : R ! E be its representation by arclength, namely jc_ðtÞj 1⁄4 1 for any t A R where R is the set of all real numbers. Shinetsu Tamura and Nobuhiro Innami space W is foliated by closed curves invariant under the billiard ball map j. Let aðxÞ denote the slope of the billiard ball trajectory determined by x for x A W. It is known that all points x which are in a j-invariant closed curve f not null-homotopic in W have the same slopes ([1], [9]). Suppose there exists a sequence of j-invariant closed curves fn not null-homotopic in W whose slopes an 1⁄4 að fnÞ converge to L=2. Let f be a j-invariant closed curve not null-homotopic in W and f À the curve consisting of the points xÀ which correspond to the reversed billiard ball trajectories to x A f. Let s 1⁄4 ðsjÞj A Z and t 1⁄4 ðtjÞj A Z be configurations for billiard ball trajectories with t1 > s1.

Preliminaries
Convex Parts of Caustics
Parallel Axiom and Periodic Trajectory
Examples
Proofs
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