Cutting sequences on square-tiled surfaces

Johnson, Charles Christopher

doi:10.1007/s10711-017-0227-z

Cited by 2 publications

(1 citation statement)

References 15 publications

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“…For example, in Section 4 we sketch a similar result for cutting sequences. [Joh17]. Beyond the Veech case, there is a long string of papers of Lopez-Narbel developing a language-theoretic formulation of generalized Sturmian sequences for interval exchange transformations, beginning with [LN95].…”

Section: Introductionmentioning

confidence: 99%

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

Duchin¹,

Erlandsson²,

Leininger³

et al. 2018

Preprint

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We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.

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Section: Introductionmentioning

confidence: 99%

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

Duchin¹,

Erlandsson²,

Leininger³

et al. 2018

Preprint

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Particle transport in open polygonal billiards: A scattering map

Orchard,

Frascoli,

Rondoni

et al. 2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years, polygonal billiards have attracted much attention due to their application in the understanding of anomalous transport, but also at the fundamental level, due to their connections with diverse fields in mathematics. We explore this complexity and its consequences on the properties of particle transport in infinitely long channels made of the repetitions of an elementary open polygonal cell. Borrowing ideas from the Zemlyakov–Katok construction, we construct an interval exchange transformation classified by the singular directions of the discontinuities of the billiard flow over the translation surface associated with the elementary cell. From this, we derive an exact expression of a scattering map of the cell connecting the outgoing flow of trajectories with the unconstrained incoming flow. The scattering map is defined over a partition of the coordinate space, characterized by different families of trajectories. Furthermore, we obtain an analytical expression for the average speed of propagation of ballistic modes, describing with high accuracy the speed of propagation of ballistic fronts appearing in the tails of the distribution of the particle displacement. The symbolic hierarchy of the trajectories forming these ballistic fronts is also discussed.

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Cutting sequences on square-tiled surfaces

Cited by 2 publications

References 15 publications

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

Particle transport in open polygonal billiards: A scattering map

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