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Ergodicity of a class of truncated elliptical billiards
Full text linkWe consider a class of billiard tables obtained by intersecting elliptical domains x2/a2 + y2 ≤ 1, a > 1 with horizontal strips |y| ≤ h < 1. The boundary of these tables consists of two elliptical arcs connected by two parallel straight segments. We prove that the billiards in these tables have non-vanishing Lyapunov exponents for h < min(1/a, 1/√2), and are ergodic for h < 1/√1 +a2
A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities
Full text linkIn this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski
Semi-focusing billiards: ergodicity
Full text linkIn Bunimovich and Del Magno [Semi-focusing billiards: hyperbolicity. Comm. Math. Phys. 262 (2006), 1732], we proved that billiards in certain three-dimensional convex domains are hyperbolic. In this paper, we continue the study of these systems, and prove that they enjoy the Bernoulli property. This result answers affirmatively a long-standing question on the existence of ergodic billiards in convex domains in dimensions greater than two. Besides, it shows that the chaotic components of the first rigorously investigated three-dimensional billiards with mixed phase space (mushroom billiards), introduced in Bunimovich and Del Magno, are in fact Bernoulli
Semi-focusing billiards: Hyperbolicity
Full text linkIn this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of R3. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of R3 with divided phase space
Singular sets of planar hyperbolic billiards are regular
Full text linkMany planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be C^2 differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class C^2 with finitely many intersection points
Track Billiards
Full text linkWe study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of 3-dimensional billiards. Interestingly, we find that for some track billiards, the mechanism generating hyperbolicity is not the defocusing one, which requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary
An infinite step billiard
Full text linkA class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain Ω = ∪n∈N[n,n + 1] × [0, p_n], with elastic reflections on the boundary; here p_0 = 1, p_n > 0 and pn ↘ 0. After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example p_n = 2^{-n}. Playing an important role in this case are the so-called escape orbits, that is, orbits going to +∞ monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard
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Asymptotic stability of the optimal filter for random chaotic maps
Full text linkThe asymptotic stability of the optimal filtering process in discrete time is revisited. The filtering process is the conditional probability of the state of a Markov process, called the signal process, given a series of observations. Asymptotic stability means that the distance between the true filtering process and a wrongly initialised filter converges to zero as time progresses. In the present setting, the signal process arises through iterating an i.i.d. sequence of uniformly expanding random maps. It is showed that for such a signal, the asymptotic stability is exponential provided that its initial conditions are sufficiently smooth. Similar to previous work on this problem, Hilbert’s projective metric on cones is employed as well as certain mixing properties of the signal, albeit with important differences. Mixing and ultimately filter stability in the present situation are due to the expanding dynamics rather than the stochasticity of the signal process. In fact, the conditions even permit iterations of a fixed (nonrandom) expanding map
DISSIPATIVE OUTER BILLIARDS: A CASE STUDY
Full text linkThis paper is dedicated to the memory of the third author, who unexpectedly passed away while the paper was being completed. Abstract. We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the as-ymptotic periodic behavior is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experiments suggesting that dissipative outer billiards about regular polygon are generically asymptotically periodic. 1. Introduction an
Non-Hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics
Full text linkWe introduce and investigate billiard systems with an adjusted ray dynamics that accounts for modifications of the conventional reflection of rays due to universal wave effects. We show that even small modifications of the specular reflection law have dramatic consequences on the phase space of classical billiards. These include the creation of regions of non-Hamiltonian dynamics, the breakdown of symmetries, and changes in the stability and morphology of periodic orbits. Focusing on optical microcavities, we show that our adjusted dynamics provides the missing ray counterpart to previously observed wave phenomena and we describe how to observe its signatures in experiments. Our findings also apply to acoustic and ultrasound waves and are important in all situations where wavelengths are comparable to system sizes, an increasingly likely situation considering the systematic reduction of the size of electronic and photonic devices
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