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Birkhoff attractors of dissipative billiards

Olga Bernardi, Anna Florio, Martin Leguil

TL;DR

The paper analyzes dissipative billiard maps in planar convex domains, introducing the Birkhoff attractor Λ_λ as the invariant, separating core of the global attractor and studying its dependence on table geometry and the dissipation strength. It proves that strong dissipation yields a normally contracted C^1 (often C^{k-1}) graph attractor, with a rotation number fixed at 1/2 in generic cases, while mild dissipation yields topologically and dynamically rich attractors that can be indecomposable and host horseshoes. For circular and elliptic tables, the authors obtain explicit descriptions: circles give the simple Λ_λ = T × {0}, and ellipses yield Λ_λ = finite unions of unstable manifolds of saddle 2-periodic orbits together with sinks, with graph structure persisting under small perturbations when e is small. The work further develops a rotation-number framework for twist maps, establishing that generic instability regions around the zero section imply a positive gap ρ^+ − ρ^−, which in turn guarantees indecomposable attractors and horseshoes, thereby linking geometric table properties to global dynamical complexity and phase transitions in Λ_λ.

Abstract

We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with "pinched" curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that generically the Birkhoff attractor is complicated, both from the topological and the dynamical point of view.

Birkhoff attractors of dissipative billiards

TL;DR

The paper analyzes dissipative billiard maps in planar convex domains, introducing the Birkhoff attractor Λ_λ as the invariant, separating core of the global attractor and studying its dependence on table geometry and the dissipation strength. It proves that strong dissipation yields a normally contracted C^1 (often C^{k-1}) graph attractor, with a rotation number fixed at 1/2 in generic cases, while mild dissipation yields topologically and dynamically rich attractors that can be indecomposable and host horseshoes. For circular and elliptic tables, the authors obtain explicit descriptions: circles give the simple Λ_λ = T × {0}, and ellipses yield Λ_λ = finite unions of unstable manifolds of saddle 2-periodic orbits together with sinks, with graph structure persisting under small perturbations when e is small. The work further develops a rotation-number framework for twist maps, establishing that generic instability regions around the zero section imply a positive gap ρ^+ − ρ^−, which in turn guarantees indecomposable attractors and horseshoes, thereby linking geometric table properties to global dynamical complexity and phase transitions in Λ_λ.

Abstract

We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with "pinched" curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that generically the Birkhoff attractor is complicated, both from the topological and the dynamical point of view.
Paper Structure (15 sections, 44 theorems, 169 equations, 10 figures)

This paper contains 15 sections, 44 theorems, 169 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Dissipative maps and Birkhoff attractors
  3. General facts about dissipative billiards
  4. Dissipative billiard map
  5. Properties of the Birkhoff attractor for axially symmetric billiards
  6. Bifurcations of $2$-periodic points
  7. Birkhoff attractor for circular and elliptic billiards
  8. Birkhoff attractors for strong dissipation
  9. Normally contracted Birkhoff attractors for strong dissipation
  10. Examples and further consequences
  11. Topologically complex Birkhoff attractors for mild dissipation
  12. Twist diffeomorphisms
  13. Upper and lower rotation numbers and Charpentier's result
  14. The dissipative billiard case
  15. Proof of Lemma \ref{['lemme vp reelles']} and Lemma \ref{['lemma noncst disp']}: bifurcation of eigenvalues at $2$-periodic points for dissipative billiard maps

Key Result

Theorem 2

Given an ellipse $\mathcal{E}$ of eccentricity $e\in (0,1)$, let $f_\lambda \colon \mathbb{A} \to f_\lambda(\mathbb{A})\subset \mathrm{int}(\mathbb A)$ be a dissipative billiard map within $\mathcal{E}$ in the sense of Definition definit diss bill (we allow non-constant dissipation). Then, the $2$-p Moreover, for $i=0,1$, $\mathcal{W}^u(f_\lambda^i(H);f_\lambda^2)\setminus \{f_\lambda^i(H)\}$ is t

Figures (10)

  • Figure 1: The standard billiard map and its dissipative counterpart.
  • Figure 2: Birkhoff attractor of a dissipative billiard map $f_\lambda$ within an ellipse of non-zero eccentricity when the dissipation is mild, i.e., $\lambda$ is close to $1$.
  • Figure 3: The geometric condition $\max_{s \in \mathbb T} \tau(s)\mathcal{K}(s)<-1$ in Definition \ref{['defi set d k']}. Here, $R(s):=-\frac{1}{\mathcal{K}(s)}$ is the radius of curvature, and $\mathscr{C}(O(s),R(s))$ is the osculating circle at $\Upsilon(s)$.
  • Figure 4: Phase transition for a $C^k$-generic domain near an ellipse of non-zero eccentricity, $k \geq 3$.
  • Figure 5: The Lyapunov function $-\zeta$.
  • ...and 5 more figures

Theorems & Definitions (121)

  • Definition 1
  • Remark 1.1
  • Theorem 2
  • Corollary 3
  • Definition 4
  • Theorem 5
  • Remark 1.3
  • Remark 1.4
  • Theorem 6
  • Remark 1.5
  • ...and 111 more