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Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent

Illya Koval

Abstract

The problem of the existence of an analytic normal form near an equilibrium point of an area-preserving map and analyticity of the associated coordinate change is a classical problem in dynamical systems going back to Poincaré and Siegel. One important class of examples of area-preserving maps consists of the collision maps for planar billiards. Recently, Treschev discovered a formal $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric billiard with locally linearizable dynamics and conjectured its convergence. Since then, a Gevrey regularity for such a billiard was proven by Wang and Zhang, but the original problem about analyticity still remains open. We extend the class of billiards by relaxing the symmetry condition and allowing conjugacies to non-linear analytic integrable normal forms. To keep the formal solution unique, odd table derivatives and the normal form are treated as parameters of the problem. We show that for the new problem, the series of the billiard table diverge for general parameters by proving the optimality of Gevrey bounds. The general parameter set is prevalent (in a certain sense has full measure) and it contains an open set. Instead of considering the problem in a functional sense and iterating approximation procedures, we employ formal power series methods and one-by-one directly reconstruct all the Taylor coefficients of the table. In order to prove that on an open set Taylor series diverges we define a Taylor recurrence operator and prove that it has a cone property. All solutions in that cone are only Gevrey regular and not analytic.

Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent

Abstract

The problem of the existence of an analytic normal form near an equilibrium point of an area-preserving map and analyticity of the associated coordinate change is a classical problem in dynamical systems going back to Poincaré and Siegel. One important class of examples of area-preserving maps consists of the collision maps for planar billiards. Recently, Treschev discovered a formal symmetric billiard with locally linearizable dynamics and conjectured its convergence. Since then, a Gevrey regularity for such a billiard was proven by Wang and Zhang, but the original problem about analyticity still remains open. We extend the class of billiards by relaxing the symmetry condition and allowing conjugacies to non-linear analytic integrable normal forms. To keep the formal solution unique, odd table derivatives and the normal form are treated as parameters of the problem. We show that for the new problem, the series of the billiard table diverge for general parameters by proving the optimality of Gevrey bounds. The general parameter set is prevalent (in a certain sense has full measure) and it contains an open set. Instead of considering the problem in a functional sense and iterating approximation procedures, we employ formal power series methods and one-by-one directly reconstruct all the Taylor coefficients of the table. In order to prove that on an open set Taylor series diverges we define a Taylor recurrence operator and prove that it has a cone property. All solutions in that cone are only Gevrey regular and not analytic.
Paper Structure (30 sections, 27 theorems, 212 equations, 9 figures)

This paper contains 30 sections, 27 theorems, 212 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Cone property for Taylor recurrence operator
  4. Formal conjugacy problem
  5. Relation to other billiard problems
  6. Standard maps and beyond
  7. Trees and formal methods
  8. Comments on the theorem assumptions
  9. Borel transform of Gevrey series
  10. Setting and plan of the proof
  11. Plan of the proof
  12. Acknowledgments
  13. Main equation and symmetries
  14. Tree method
  15. Motivation
  16. ...and 15 more sections

Key Result

Theorem 1

If $q_{odd}(t)$ and $b(r^2)$ are real analytic functions and $b_0/\pi$ has restricted partial quotients, then GTPS is $1+1$-Gevrey: for some $C \in \mathbb{R}$.

Figures (9)

  • Figure 1: Treschev's domain, plotted using its first $25$ power series terms. An orbit coming from the formal conjugacy is also plotted, its error is at most $10^{-37}$. Despite the shape looking like an ellipse, the fourth derivative of the pictured $q(t)$ at $0$ differs from that of the closest ellipse by more than $1$.
  • Figure 2: Cone property of operator $T$. The dynamics of the point under $T$ is shown with gray arrows and purple arrows denote the errors of approximation.
  • Figure 3: An example of a formal tree.
  • Figure 4: A unique reconstruction of elements of a contributing tree from Figure \ref{['fig1']}. For $2$ vertices, their parameters are stated.
  • Figure 5: An example of a tree, principal at several vertices.
  • ...and 4 more figures

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Theorem 4
  • Conjecture 1.1
  • Remark 4
  • Remark 5
  • ...and 57 more