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May 1, 2026Annals of MathematicsOpen Access

Weakly mixing polygonal billiards

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Authors

JCJon ChaikaGFGiovanni Forni

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Overview

Mathematical analysis demonstrates weakly mixing billiard flows across a residual set of polygons, indicating typical chaotic dynamics in these systems.

Key Points

  • To determine whether billiard flows on generic non-rational polygons exhibit the dynamical property of weak mixing with respect to the Liouville measure.
  • Applied a Baire category argument to connect properties of translation surfaces with polygonal billiard flows.
  • Analyzed the spectrum of directional flows on arbitrary translation surfaces to check for shared non-trivial eigenvalues.
  • Demonstrated that a residual set of non-rational polygons produces billiard flows that are weakly mixing with respect to Liouville measure on the unit tangent bundle.
  • Proved that directional flows on any translation surface lack non-trivial common eigenvalues for almost all pairs of directions, guaranteeing ergodicity for their product flow under Lebesgue measure.

Cite This Study

Chaika et al. (2026) studied this question.

synapsesocial.com/papers/6a7a003aa861e9f5993263cehttps://doi.org/10.4007/annals.2026.203.3.1
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1Random Circular Billiards on Surfaces of Constant Curvature: Pseudo Integrability and Mixing2024
  2. 2On rationally integrable planar dual multibilliards and piecewise smooth projective billiards2024 · 1 citations
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  5. 5Rapid mixing for compact group extensions of hyperbolic flows2024