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Two billiard domains whose billiard maps are Lazutkin conjugates are the same

Corentin Fierobe

TL;DR

The paper tackles whether two strictly convex planar billiards with identical Lazutkin-coordinate expressions for their billiard maps must be isometric. It develops the Lazutkin-coordinate framework, deriving explicit expansions for the Lazutkin map with coefficients $\alpha_3$, $\alpha_4$, and $\beta_4$ expressed in terms of the radius of curvature $\varrho$ and its derivatives. It argues that equality of these Lazutkin maps forces the radii of curvature to coincide up to a reparameterization, leading to isometry, and extends the argument to general diffeomorphisms via a conjugacy analysis. However, the paper acknowledges that the results rely on flawed expansions in Lazutkin's book, which undermines the claimed conclusions about isometry and conjugacy.

Abstract

This paper aimed to show that two billiards whose billiard maps share the same expression in Lazutkin coordinates are isometric. However, the results are incorrect as they rely on erroneous computations in Lazutkin's book concerning the expansion of the billiard map in Lazutkin coordinates.

Two billiard domains whose billiard maps are Lazutkin conjugates are the same

TL;DR

The paper tackles whether two strictly convex planar billiards with identical Lazutkin-coordinate expressions for their billiard maps must be isometric. It develops the Lazutkin-coordinate framework, deriving explicit expansions for the Lazutkin map with coefficients , , and expressed in terms of the radius of curvature and its derivatives. It argues that equality of these Lazutkin maps forces the radii of curvature to coincide up to a reparameterization, leading to isometry, and extends the argument to general diffeomorphisms via a conjugacy analysis. However, the paper acknowledges that the results rely on flawed expansions in Lazutkin's book, which undermines the claimed conclusions about isometry and conjugacy.

Abstract

This paper aimed to show that two billiards whose billiard maps share the same expression in Lazutkin coordinates are isometric. However, the results are incorrect as they rely on erroneous computations in Lazutkin's book concerning the expansion of the billiard map in Lazutkin coordinates.

Paper Structure

This paper contains 4 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Lazutkin coordinates
  3. Domains with the same Lazutkin billiard maps
  4. Diffeomorphisms conjugating billiard maps

Key Result

Theorem 1

Assume that two domains $\Omega_1$ and $\Omega_2$ with $\mathscr C^6$-smooth boundaries have the same billiard map in Lazutkin coordinates, namely $T_1^L=T_2^L$. Then $\Omega_1$ and $\Omega_2$ are isometric.

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Proposition 4
  • Remark 5
  • proof
  • Proposition 6
  • proof
  • Remark 7