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On infinitely many foliations by caustics in strictly convex open billiards

Alexey Glutsyuk

Abstract

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve $C$ whose tangent lines are reflected by the billiard to lines tangent to $C$. The famous Birkhoff Conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin's theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve $γ$. We prove that there exists a domain $U$ adjacent to $γ$ from the convex side and a $C^\infty$-smooth foliation of $U\cupγ$ whose leaves are $γ$ and (non-closed) caustics of the billiard. This generalizes a previous result by R.Melrose, which yields existence of a germ of foliation as above at a boundary point. We show that there exists a continuum of above foliations by caustics whose germs at each point in $γ$ are pairwise different. We prove a more general version of this statement in the cases, when $γ$ is just an arc, and also when both $γ$ and the caustics are immersed curves. It also applies to a billiard bounded by a closed strictly convex curve $γ$ and yields infinitely many "immersed" foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty}$-lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

On infinitely many foliations by caustics in strictly convex open billiards

Abstract

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve whose tangent lines are reflected by the billiard to lines tangent to . The famous Birkhoff Conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin's theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve . We prove that there exists a domain adjacent to from the convex side and a -smooth foliation of whose leaves are and (non-closed) caustics of the billiard. This generalizes a previous result by R.Melrose, which yields existence of a germ of foliation as above at a boundary point. We show that there exists a continuum of above foliations by caustics whose germs at each point in are pairwise different. We prove a more general version of this statement in the cases, when is just an arc, and also when both and the caustics are immersed curves. It also applies to a billiard bounded by a closed strictly convex curve and yields infinitely many "immersed" foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called -lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

Paper Structure

This paper contains 20 sections, 41 theorems, 143 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Main result: an open convex arc has infinitely many foliations by caustics
  3. Background material: symplectic properties of billiard ball map
  4. Generalization to $C^{\infty}$-lifted strongly billiard-like maps
  5. Unique determination of jets. Space of germs of foliations. Non-uniqueness
  6. Corollaries on conjugacy of open billiard maps near the boundary
  7. Plan of the proof of main results
  8. Historical remarks
  9. Construction of foliation by invariant curves. Proofs of main results
  10. Marvizi--Melrose construction of an "up-to-flat" first integral
  11. Step 1. Construction of an invariant function on a neighborhood of fundamental domain
  12. Step 2. Extension by dynamics
  13. Step 3. Regularity and flatness. End of proof of Theorem \ref{['thm33']}, Statement 1)
  14. Normal form. Proof of Statement 2) of Theorem \ref{['thm33']}
  15. Proof of existence in Theorem \ref{['thm3']}. Proof of Theorem \ref{['addthm3']}
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

1) Consider an open billiard bounded by a strictly convex $C^{\infty}$-smooth curve $\gamma\subset\mathbb R^2$: a one-dimensional submanifold parametrized by interval. There exists a simply connected domain $U$ adjacent to $\gamma$ from the convex side that admits a foliation by caustics of the bill

Figures (5)

  • Figure 1: The billiard ball map and a caustic.
  • Figure 2: A Birkhoff integrable billiard.
  • Figure 3: An open strictly convex planar billiard and its caustics. Here the ambient plane $\mathbb R^2$ is presented together with its boundary: the infinity line.
  • Figure 4: An immersed foliation by immersed caustics.
  • Figure 5: The fundamental domain $\Delta$ and its sectorial neighborhood $S_{\chi,\eta}$.

Theorems & Definitions (65)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
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  • Definition \oldthetheorem
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  • ...and 55 more