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The billiard inside an ellipse deformed by the curvature flow


Authors: Josué Damasceno, Mario J. Dias Carneiro and Rafael Ramírez-Ros
Journal: Proc. Amer. Math. Soc. 145 (2017), 705-719
MSC (2010): Primary 37E40, 37J45, 37B40, 53C44
DOI: https://doi.org/10.1090/proc/13351
Published electronically: September 29, 2016
MathSciNet review: 3577872
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Abstract | References | Similar Articles | Additional Information

Abstract: The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the integrability, increases the topological entropy, splits the separatrices in a transverse way, and breaks all resonant convex caustics.


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Additional Information

Josué Damasceno
Affiliation: Departamento de Matemática, Universidade Federal de Ouro Preto, 35.400–000, Ouro Preto, Brazil
Email: josue@iceb.ufop.br

Mario J. Dias Carneiro
Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123–970, Belo Horizonte, Brazil
Email: carneiro@mat.ufmg.br

Rafael Ramírez-Ros
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
Email: rafael.ramirez@upc.edu

DOI: https://doi.org/10.1090/proc/13351
Keywords: Billiard, curvature flow, topological entropy, Melnikov method
Received by editor(s): November 4, 2015
Received by editor(s) in revised form: April 6, 2016
Published electronically: September 29, 2016
Additional Notes: The third author was supported in part by CUR-DIUE Grant 2014SGR504 (Catalonia) and MINECO-FEDER Grant MTM2015-65715-P (Spain).
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society

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