# Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards

Bálint P, De Simoi J, Kaloshin V, Leguil M. 2019. Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards. Communications in Mathematical Physics. 374(3), 1531–1575.

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Author

Bálint, Péter;
De Simoi, Jacopo;
Kaloshin, Vadim

^{ISTA}^{}; Leguil, MartinAbstract

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.

Publishing Year

Date Published

2019-05-09

Journal Title

Communications in Mathematical Physics

Volume

374

Issue

3

Page

1531-1575

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### Cite this

Bálint P, De Simoi J, Kaloshin V, Leguil M. Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards.

*Communications in Mathematical Physics*. 2019;374(3):1531-1575. doi:10.1007/s00220-019-03448-xBálint, P., De Simoi, J., Kaloshin, V., & Leguil, M. (2019). Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards.

*Communications in Mathematical Physics*. Springer Nature. https://doi.org/10.1007/s00220-019-03448-xBálint, Péter, Jacopo De Simoi, Vadim Kaloshin, and Martin Leguil. “Marked Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards.”

*Communications in Mathematical Physics*. Springer Nature, 2019. https://doi.org/10.1007/s00220-019-03448-x.P. Bálint, J. De Simoi, V. Kaloshin, and M. Leguil, “Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards,”

*Communications in Mathematical Physics*, vol. 374, no. 3. Springer Nature, pp. 1531–1575, 2019.Bálint, Péter, et al. “Marked Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards.”

*Communications in Mathematical Physics*, vol. 374, no. 3, Springer Nature, 2019, pp. 1531–75, doi:10.1007/s00220-019-03448-x.**All files available under the following license(s):**

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arXiv 1809.08947