Ergodicity of Billiard Flows and Quadratic Differentials

Kerckhoff, Steven P.; Masur, Howard; Smillie, John

doi:10.2307/1971280

Cited by 252 publications

(208 citation statements)

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“…Theorem 4.7 (Kerkhoff-Masur-Smillie [KMS86]). For every (X, ω) and almost every θ ∈ [0, 2π), g t (r θ (X, ω)) is recurrent (as θ is fixed and t → ∞).…”

Section: The Straight Line Flowmentioning

confidence: 99%

Translation surfaces and their orbit closures: An introduction for a broad audience

Wright¹

2015

EMS Surv. Math. Sci.

107

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Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2, R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming diverse areas of mathematics.Many areas of mathematics, from algebraic geometry and number theory, to dynamics and topology, can be brought to bear on this topic, and furthermore known examples of orbit closures are interesting from all these points of view.This survey is an invitation for mathematicians from different backgrounds to become familiar with the subject. Little background knowledge, beyond the definition of a Riemann surface and its cotangent bundle, is assumed, and top priority is given to presenting a view of the subject that is at once accessible and connected to many areas of mathematics.Acknowledgements: I am especially grateful to Ronen Mukamel for helpful conversations that shaped the presentation of real multiplication in genus 2, and to Curtis McMullen for helpful comments and corrections. I am also very grateful to

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“…Theorem 4.7 (Kerkhoff-Masur-Smillie [KMS86]). For every (X, ω) and almost every θ ∈ [0, 2π), g t (r θ (X, ω)) is recurrent (as θ is fixed and t → ∞).…”

Section: The Straight Line Flowmentioning

confidence: 99%

Translation surfaces and their orbit closures: An introduction for a broad audience

Wright¹

2015

EMS Surv. Math. Sci.

107

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“…Note that strictly convex billiards whose boundary is C 6 are not ergodic (see [15]), and there are sequences of positive density of eigenfunctions which concentrate on caustics. Rational polygonal billiards are not ergodic, but generic polygonal billiards are ergodic (see [13]). Recently, Hassell has shown in [9] that there exist some convex (stadium-shaped) sets satisfying QE but not QUE.…”

Section: Second Problem and Quantum Ergodicitymentioning

confidence: 99%

“…We end the study of the second problem by considering the spectral truncation of the functional J defined by (13) J N (χ ω ) = min…”

Section: Second Problem and Quantum Ergodicitymentioning

confidence: 99%

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Privat¹,

Trélat²,

Zuazua³

2013

Journées Équations Aux Dérivées Partielles

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International audienceThis paper is a proceedings version of an ongoing work, and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega\subset\R^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega$ of Lebesgue measure $\vert\omega\vert=L\vert\Omega\vert$, where $L\in(0,1)$ is fixed. We denote by $\mathcal{U}_L$ the class of all possible such subsets. Let $T>0$. We consider first the benchmark problem of maximizing the observability energy $\int_0^T\int_\omega\vert y(t,x)^2\, dx\, dt$ over $\mathcal{U}_L$, for fixed initial data. There exists at least one optimal set and we provide some results on its regularity properties. In view of practical issues, it is far more interesting to consider then the problem of maximizing the observability constant. But this problem is difficult and we propose a slightly different approach which is actually more relevant for applications. We define the notion of a randomized observability constant, where this constant is defined as an averaged over all possible randomized initial data. This constant appears as a spectral functional which is an eigenfunction concentration criterion. It can be also interpreted as a time asymptotic observability constant. This maximization problem happens to be intimately related with the ergodicity properties of the domain $\Omega$. We are able to compute the optimal value under strong ergodicity properties on $\Omega$ (namely, Quantum Unique Ergodicity). We then provide comments on ergodicity issues, on the existence of an optimal set, and on spectral approximations

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“…The proof of Theorem A.4 hinges on the following two basic geometric lemmas which are pretty simple to prove (see [KMS,§3], [V2] or [MW]) Proposition A.5 (Cf. [MW,Prop.…”

Section: A2 Proof Of Theorems A1 For Surfaces With No Puncturesmentioning

confidence: 99%