2024/10/17 by Corentin Fierobe, Fierobe, Corentin, Alfonso Sorrentino +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2410.13777
openalex publication_date 2024/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space.