Strictly convex ergodic billiards
Mathematics Research Reports, Volume 7 (2026), pp. 71-79

We produce the first known strictly convex ergodic billiards: the intersection of two circles of which one contains the centers of both.

Received:
Published online:
DOI: 10.5802/mrr.28
Classification: 37C83
Keywords: Hyperbolicity, billiards, ergodicity

Wentao Fan  1 ; Boris Hasselblatt  1

1 Department of Mathematics, Tufts University, Medford, MA 02155
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Wentao Fan; Boris Hasselblatt. Strictly convex ergodic billiards. Mathematics Research Reports, Volume 7 (2026), pp. 71-79. doi: 10.5802/mrr.28
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[1] G. Benettin; J. -M. Strelcyn Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy, Phys. Rev. A, Volume 17 (1978), pp. 773-785 https://link.aps.org/doi/10.1103/PhysRevA.17.773 | DOI

[2] Giancarlo Benettin; Luigi Galgani; Jean-Marie Strelcyn Kolmogorov entropy and numerical experiments, Phys. Rev. A, Volume 14 (1976), pp. 2338-2345 https://link.aps.org/doi/10.1103/PhysRevA.14.2338 | DOI

[3] L. A. Bunimovich The ergodic properties of certain billiards, Funkcional. Anal. i Priložen., Volume 8 (1974) no. 3, pp. 73-74 | MR

[4] L. A. Bunimovich On absolutely focusing mirrors, Ergodic theory and related topics, III (Güstrow, 1990) (Lecture Notes in Math.), Volume 1514, Springer, Berlin, 1992, pp. 62-82 | DOI | MR | Zbl

[5] Leonid Bunimovich; Hong-Kun Zhang; Pengfei Zhang On another edge of defocusing: hyperbolicity of asymmetric lemon billiards, Comm. Math. Phys., Volume 341 (2016) no. 3, pp. 781-803 | DOI | MR | Zbl

[6] Leonid A. Bunimovich; Luz V. Vela-Arevalo Some new surprises in chaos, Chaos, Volume 25 (2015) no. 9, p. 097614, 11 | DOI | MR

[7] Keith Burns; Amie Wilkinson On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), Volume 171 (2010) no. 1, pp. 451-489 | DOI | MR | Zbl

[8] Jingyu Chen; Luke Mohr; Hong-Kun Zhang; Pengfei Zhang Ergodicity of the generalized lemon billiards, Chaos, Volume 23 (2013) no. 4, p. 043137, 12 | DOI | MR | Zbl

[9] Nikolai Chernov; Roberto Markarian Chaotic billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006, xii+316 pages | DOI | MR | Zbl

[10] Yves Coudène Théorie ergodique et systèmes dynamiques, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)], EDP Sciences, Les Ulis; CNRS Éditions, Paris, 2012, vi+198 pages | MR | Zbl

[11] Gianluigi Del Magno; Roberto Markarian A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities, Ergodic Theory Dynam. Systems, Volume 33 (2013) no. 4, pp. 983-1007 | DOI | MR | Zbl

[12] Victor J. Donnay Using integrability to produce chaos: billiards with positive entropy, Comm. Math. Phys., Volume 141 (1991) no. 2, pp. 225-257 http://projecteuclid.org/euclid.cmp/1104248299 | MR | DOI | Zbl

[13] Wentao Fan; Boris Hasselblatt Strictly convex ergodic billiards, preprint (2026)

[14] Wentao Fan; Boris Hasselblatt Uniform Hyperbolicity of strictly convex billiards, preprint (2026)

[15] Todd Fisher; Boris Hasselblatt Hyperbolic flows, Zurich Lectures in Advanced Mathematics, EMS Publishing House, Berlin, [2019] ©2019, xiv+723 pages | DOI | MR

[16] Giovanni Forni Existence of a Periodic Orbit for Billiards in Polygons, 2026 | arXiv | Zbl

[17] Boris Hasselblatt Anatole Katok, Ergodic Theory Dynam. Systems, Volume 42 (2022) no. 2, pp. 321-388 | DOI | MR | Zbl

[18] Eric J. Heller; Steven Tomsovic Postmodern Quantum Mechanics, Physics Today, Volume 46 (1993) no. 7, pp. 38-46 | DOI | arXiv

[19] M. Hénon; J. Wisdom The Benettin-Strelcyn oval billiard revisited, Phys. D, Volume 8 (1983) no. 1-2, pp. 157-169 | DOI | MR | Zbl

[20] Eberhard Hopf Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl., Volume 91 (1939), pp. 261-304 | MR | Zbl

[21] Xin Jin; Pengfei Zhang Hyperbolicity of asymmetric lemon billiards, Nonlinearity, Volume 34 (2021) no. 1, pp. 92-117 | DOI | MR | Zbl

[22] A. B. Katok Billiard table as a playground for a mathematician, The collected works of Anatole Katok. Vol. 1, World Sci. Publishing, Singapore, [2024] ©2024, pp. 1067-1088 (Reprint of [2166929]) | MR

[23] Anatole Katok; Jean-Marie Strelcyn; F. Ledrappier; F. Przytycki Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Mathematics, 1222, Springer-Verlag, Berlin, 1986, viii+283 pages | DOI | MR | Zbl

[24] Roberto Markarian Billiards with Pesin region of measure one, Comm. Math. Phys., Volume 118 (1988) no. 1, pp. 87-97 http://projecteuclid.org/euclid.cmp/1104161909 | MR | DOI | Zbl

[25] Cesar E. Silva What is an ergodic transformation?, Notices Amer. Math. Soc., Volume 63 (2016) no. 1, pp. 26-27 | DOI | MR | Zbl

[26] Yakov Sinai What is a billiard?, Notices Amer. Math. Soc., Volume 51 (2004) no. 4, pp. 412-413 | MR | Zbl

[27] Serge Tabachnikov Geometry and billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005, xii+176 pages | DOI | MR | Zbl

[28] Maciej Wojtkowski Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems, Volume 5 (1985) no. 1, pp. 145-161 | DOI | MR | Zbl

[29] Maciej Wojtkowski Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys., Volume 105 (1986) no. 3, pp. 391-414 http://projecteuclid.org/euclid.cmp/1104115431 | MR | DOI | Zbl

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