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Let f : [0, +∞) −→ (0, +∞) be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain Q delimited by the positive x-semiaxis, the positive y-semiaxis, and the graph of f .
Under certain conditions on f , we prove that the billiard flow in Q has a hyperbolic structure and, for some examples, that it is also ergodic....
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Citations
... Initially unaware of Weiss' insight, King gave a direct 'bare-hands' proof that only a straight shot goes to infinity if the bounding function is convex. We note that this billiard with y = 1 as the third wall is ergodic [15] (likewise with a finite cusp [21]) and that cusps have also been of interest with respect to correlation decay [1-3, 5, 6, 10-12, 18,19,24]. Indeed, the main purpose of King's article was to bring measure theory to bear on this and related issues. ...
... But this holds due to (15) and (17). This completes the proof of lemma 12. ...
We consider billiards with cusps and with gravity pulling the particle into the cusp. We discover an adiabatic invariant in this context; it turns out that the invariant is in form almost identical to the Clairaut integral (angular momentum) for surfaces of revolution. We also approximate the bouncing motion of a particle near a cusp by smooth motion governed by a differential equation—which turns out to be identical to the differential equation governing geodesic motion on a surface of revolution. We also show that even in the presence of gravity pulling into a cusp of a billiard table, only the direct-hit orbit reaches the tip of the cusp. Finally, we provide an estimate of the maximal depth to which a particle penetrates the cusp before being ejected from it.
... Moreover, our phase space M is non-compact, and the smooth invariant measure for F is only σ-finite. Ergodicity of systems with singularities, preserving a smooth infinite measure is discussed for example in [39,31,32]. However, our system is significantly more complicated as we explain below. ...
... where * denotes the sum over components with |W j1 | α ≥ η. 30 Lemma 5.12 implies a uniform upper bound, and exchanging the roles of W 1 and W 2 yields the desired lower bound 31 Recall that shortened H-components were defined in the proof of the Growth Lemma 7.2 32 Recall that admissible curves have bounded Euclidean length, hence they have bounded α-length by Proposition 4.15(a) By using Lebesgue Density Theorem and Severini-Egoroff Theorem, we can conclude that, for large enough n > 0 * ...
We study a natural class of Fermi-Ulam Models that features good hyperbolicity properties and that we call dispersing Fermi-Ulam models. Using tools inspired by the theory of hyperbolic billiards we prove, under very mild complexity assumptions, a Growth Lemma for our systems. This allows us to obtain ergodicity of dispersing Fermi-Ulam Models. It follows that almost every orbit of such systems is oscillatory.
... In a seminal work, Sinai [22] proved that the billiard map of a system in a two-dimensional torus with finitely many convex obstacles is a K-automorphism. For billiards with non-compact cusps, that generate a dynamical system with an infinite invariant measure, in [15] Lenci proved an extension of the results of Katok and Strelcyn [11] for the infinite measure case and, as an application, he showed that certain tables with non-compact cusps have hyperbolic structure, that is, existence of absolutely continuous local stable and unstable manifolds. Furthermore, adapting arguments contained in [17], Lenci proved that these billiards maps are ergodic. ...
... One can see that x t is the x-coordinate of the tangent point on U. In [15], Lenci studied tables with f : IR + 0 → IR + satisfying the following assump- tions (H1) f ′′ (x) → 0 as x → +∞; (H2) |f ′ (x t )| << |f ′ (x)|; ...
... Following [15], choosing as cross-section the rebounds against the dispersing part U. we parametrize these line elements as z = (r, ϕ), r ∈ (−∞, 0] is the arc length variable along U (with r = 0 for the vertex V ) and ϕ ∈ [−π/2, π/2] is the angle between the velocity vector and the normal at the point of collision, as inFigure 3. We define the manifold M = (−∞, 0) × (−π/2, π/2) and the return map T defined on M, preserving the measure dµ = cos ϕdrdϕ. We do not define T on those points that hit tangentially U or that would end up in the vertex V . ...
Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing proposed by Krengel and Sucheston for systems with invariant infinite measure, to show that the billiard whose table is constituted by the x-axis and and the portion in the plane below the graph of $f(x)=\frac{1}{x+1}$ is mixing and the speed of mixing is polynomial. Comment: 31 pages and 7 figures
... A common approach in the classical setting is via dynamical systems for which the reflection rule is elastic. In this setting, we mention that billiards in certain unbounded domains resembling the tubes considered here (at least for γ ≤ 0) have been studied; see for instance [8,202122 and references therein. An infinite-tube billiard model with a stochastic component (cf the γ = 0 case here) is analyzed in [1]. ...
We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion
with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with
ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process
into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle,
including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process
to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure
bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also
of some independent interest.
... However, hyperbolic and expanding maps with infinite invariant measure appear, more and more often, in various applications. Recently Lenci [55], [56] extended Pesin theory and Sinai's (fundamental) ergodic theorem to unbounded dispersing billiard tables (regions under the graph of a positive monotonically decreasing function y = f (x) for 0 ≤ x < ∞), where the collision map, and often the flow as well, have infinite invariant measures. Another example that we already mentioned is the periodic Lorentz gas with a diffusive particle, but this one can be reduced, because of its symmetries, to a finite measure system by factoring out the Z 2 action (Section 2). ...
Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models.
... Our construction is completely independent on the choice of f on the interval (0, a 1 ), and one may use this additional freedom to tune f on (0, a 1 ) in such a way that the billiard flow on D is ergodic. It seems plausible that this is the case if the billiard flow on the restricted compact region D 0 = {(x, y) ∈ R 2 : 0 < x < a 1 , 0 < y < f (x)} is ergodic (as in the examples displayed in figs. 1 and 2), but to the best of my knowledge there are no rigorous results in this direction (see however [23,24,16] for proofs of ergodicity for different classes of non-compact domains). A further interesting class of examples are infinite pseudo-integrable billiards ( fig. 3) that are known to be ergodic 5 for almost all initial directions [12]. ...
We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.
... This note is a follow-up to [Le2], where we studied a certain family of billiards in the plane. A billiard is a dynamical system defined by the free motion of a material point inside a domain, called the table, with the prescription that when the point hits the boundary of the table it gets reflected at an angle equal to the angle of incidence. ...
... The family introduced in [Le2] was defined as follows: To a three-times differentiable function f : [0, +∞) −→ (0, +∞), convex, vanishing at +∞ (and thus bounded), we associate the table Q delimited by the positive x-semiaxis, the positive y-semiaxis, and the graph of f , as in Fig. 1. ...
... From a more mathematical standpoint the relevance of our billiards lies in the fact that they give rise to infinite-measure hyperbolic dynamical systems, about which not much is known. We do not delve in this issue here, as a good part of the introduction of [Le2] is devoted to it. Let us just mention that, if one is interested in studying the stochastic properties of these systems (e.g., ergodicity, as in our case), one cannot apply the fundamental results of [KS], where a version of Pesin's theory is engineered to be used on finite-measure hyperbolic systems with singularities. ...
In Lenci 2002 the following class of billiards was studied: For f:0,)→(0,) convex, sufficiently smooth, and vanishing at infinity, let the billiard table be defined by Q, the planar domain delimited by the positive x semiaxis, the positive y semiaxis, and the graph of f . For a large class of f we proved that the billiard map was hyperbolic. Furthermore we gave an example of a family of f that makes this map ergodic. Here we extend the latter result to a much wider class of functions.
... (It might also happen that ∂W u (x) intersects the closure of S − ∞ -which is typically the whole M-but standard reasonings show that this cannot occur for more than a null-measure set of x's. See [L,Rk. 8.9].) ...
We prove that any generic (i.e., possibly aperiodic) Lorenz gas in two dimensions, with nite horizon and non-degenerate geometrical features, is ergodic if it is recurrent. We also give examples of aperiodic recurrent gases.
... Bilhares com cúspides, ou seja, quinas comângulo zero, foram estudados, por exemplo, por Leontovich [18], King [12] e Lenci [17]. Além disso, versões poligonais desse mesmo sistema foram apresentadas por Degli Esposti, Del Magno e Lenci ([6] e [7]). ...
... Este resultado foi obtido por Lenci em [17]. O trabalho está organizado como descrito a seguir. ...
... Este método foi apresentado em [19] para sistemas hamiltonianos com medida finita. A sua adaptação para sistemas com medida infinita foi obtida por Lenci [17] e está concentrada no Lema 9.4, denominado "Tail Bound Lemma". ...
We recapitulate results from the infinite ergodic theory that are relevant to the theory of non-extensive entropies. In particular, we recall that the Lyapunov exponent of the corresponding systems is zero and that the deviation between neighboring trajectories does not necessarily grow polynomially. Nonetheless, as we show, no single quantity can describe this subexponential growth, the generalized q-exponential expq
being, in particular, ruled out. We also revisit a number of dynamical systems preserving nonfinite ergodic measure.
