Trees and flowers on a billiard table

Abstract : In this work we study the dynamics of triangle tiling billiards. We unite geometric and combinatorial approaches in order to prove several conjectures. In particular, we prove the Tree Conjecture and the 4n+2 Conjecture, both stated by Baird-Smith, Davis, Fromm and Iyer. Moreover, we study the set of exceptional trajectories which is closely related to the set of minimal Arnoux-Rauzy maps and prove that all of such trajectories pass by all tiles. Finally, we prove that the arithmetic orbits of the Arnoux-Yoccoz map converge, up to rescaling, to the Rauzy fractal, as conjectured by Hooper and Weiss.
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https://hal.archives-ouvertes.fr/hal-02169195
Contributor : Olga Romaskevich <>
Submitted on : Tuesday, October 8, 2019 - 3:37:22 PM
Last modification on : Friday, October 11, 2019 - 1:15:23 AM

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  • HAL Id : hal-02169195, version 3

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Olga Paris-Romaskevich. Trees and flowers on a billiard table. 2019. ⟨hal-02169195v3⟩

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