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This paper proposes a dissipative variant of symplectic billiards in convex domains and studies the properties of the Birkhoff attractor.

July 30, 2026Journal of Dynamics and Differential Equations0 citationsOpen Access

Birkhoff Attractors for Dissipative Symplectic Billiards

LBLuca Baracco OBOlga Bernardi AFAnna Florio

Key Points

This paper proposes a dissipative variant of symplectic billiards in convex domains and studies the properties of the Birkhoff attractor.
Analyzed the dissipative billiard map within strictly convex planar domains.
Investigated effects of varying dissipation rates on the geometry of the Birkhoff attractor.
Compared findings with existing literature on dissipative Birkhoff billiards.
For strong dissipation, the Birkhoff attractor is identified as a normally contracted graph over the zero section.
For mild dissipation, the attractor in centrally symmetric domains is an indecomposable continuum with positive topological entropy.

Abstract

Abstract The aim of the present paper is to propose and study a dissipative variant of symplectic billiards within planar strictly convex domains. The associated billiard map is dissipative, thus it admits a compact invariant set, the so-called Birkhoff attractor. Its complexity depends on the rate of dissipation as well as on the geometry of the billiard table. We prove that ( a ) for strong dissipation, the Birkhoff attractor is a normally contracted graph over the zero section; ( b ) for mild dissipation, the Birkhoff attractor within a centrally symmetric domain is an indecomposable continuum whose restricted dynamics has positive topological entropy. We compare these results with those for dissipative Birkhoff billiards, studied in 15.

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