Strictly convex ergodic billiards
Wentao Fan, Boris Hasselblatt · arXiv: 2610.28247v1 · Summary page
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Orderly boundaries, thoroughly mixing billiards
A billiard ball following ideal reflections can generate dynamics far more complicated than its simple rules suggest. A new preprint presents the first known strictly convex ergodic billiards: sufficiently asymmetric lemon-shaped tables formed by intersecting circles. Ergodicity means, roughly, that a typical trajectory explores the available phase space according to its natural statistical distribution, rather than remaining trapped in a separate dynamical region.
The authors build on an earlier proof of uniformly hyperbolic returns to a suitable section of the motion. The new ingredient controls singularity curves, where the billiard dynamics become delicate. This establishes not only ergodicity but the stronger Bernoulli property, a precise form of statistical randomness. The result gives a concrete setting in which deterministic mechanics produces robust statistical behavior.
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