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Yet another characterizatione of Birkhoff Billiards inside discs

Klaudiusz Czudek, Jacopo De Simoi, Andrew Gad, Marco Poon

Abstract

In this short paper, we show a characterization of Birkhoff Billiards inside discs which is related to the expansion of the formal Lazutkin conjugacy at the boundary.

Yet another characterizatione of Birkhoff Billiards inside discs

Abstract

In this short paper, we show a characterization of Birkhoff Billiards inside discs which is related to the expansion of the formal Lazutkin conjugacy at the boundary.
Paper Structure (10 sections, 6 theorems, 54 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction and statement of our main results
  2. Comments and ideas for future developments
  3. The main calculation
  4. Angular functions
  5. Relation between angular functions and high-order Lazutkin coordinates
  6. Proof of the Main Theorem
  7. Solving the differential equation \ref{['eq:alpha-ODE']}
  8. Characterizing the disc
  9. Computing $X_{1}$
  10. Derivatives

Key Result

Lemma 1.1

Assume $\partial\Omega$ is smooth, for any $k > 0$ there exists a neighbourhood $U$ of $\{\varphi = 0\}$ and a diffeomorphism $\Psi:U\to V$, where $V\subset\mathbb{T}\times[0,\varepsilon)$ is a neighbourhood of $\mathbb{T}\times\{0\}$ which conjugates $f$ with the normal form We call the above normal form Lazutkin normal form of order $k$. Moreover, we can write $\Psi = (X,Y)$, where Finally, if

Figures (2)

  • Figure 1: The triple $(z_{-}, z_{0}, z_{+})$ identifies a billiard $\varepsilon$-pseudocollision if $|\varphi^{+}-\varphi^{-}| < \varepsilon$
  • Figure 2: Curves $\Gamma$ corresponding to a full period of the solution $\zeta$ with $E = 0.005$, $E = 0.02$ and $E = 0.05$ respectively (left to right). The curves appear not to be periodic, and to complete an angle of $4\pi$.

Theorems & Definitions (21)

  • Lemma 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6: On Lazutkin conjugacy equivalence
  • Remark 1.7: Distribution of collision points of periodic orbits
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2: Angular function
  • ...and 11 more