Hyperbolicity and abundance of elliptical islands in annular billiards

We study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, a two-parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdo...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 43; no. 11; pp. 3545 - 3577
Main Authors BATISTA, REGINALDO BRAZ, DIAS CARNEIRO, MÁRIO JORGE, OLIFFSON KAMPHORST, SYLVIE
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.11.2023
Subjects
Barriers
Billiards
Islands
Metric space
Original Article
Parameters
37C83
hyperbolic sets
homoclinic tangencies
billiards
37D05
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Summary:We study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, a two-parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdorff distance, as the radius goes to zero and the center of the obstacle approximates the outer boundary. The dynamics on each of these sets is conjugate to a shift with an increasing number of symbols. We also show that for many parameters, the system presents quadratic homoclinic tangencies whose bifurcation gives rise to elliptical islands (conservative Newhouse phenomenon). Thus, for many parameters, we obtain the coexistence of a ‘large’ hyperbolic set with many elliptical islands.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2022.80