TY - JOUR AB - We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table. AU - De Simoi, Jacopo AU - Kaloshin, Vadim AU - Leguil, Martin ID - 12877 JF - Inventiones Mathematicae SN - 0020-9910 TI - Marked Length Spectral determination of analytic chaotic billiards with axial symmetries VL - 233 ER - TY - JOUR AB - In the paper, we establish Squash Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia whose convex arcs are homothetic. We also establish Stadium Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls’ Barrier function from the maximal marked length spectrum associated to the rotation number 2n/4n+1. AU - Chen, Jianyu AU - Kaloshin, Vadim AU - Zhang, Hong Kun ID - 14427 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Length spectrum rigidity for piecewise analytic Bunimovich billiards ER - TY - JOUR AB - In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In contrast, we prove that any two elliptic billiard maps are C0-conjugate near their respective boundaries, and C∞-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar. AU - Koudjinan, Edmond AU - Kaloshin, Vadim ID - 12145 IS - 6 JF - Regular and Chaotic Dynamics KW - Mechanical Engineering KW - Applied Mathematics KW - Mathematical Physics KW - Modeling and Simulation KW - Statistical and Nonlinear Physics KW - Mathematics (miscellaneous) SN - 1560-3547 TI - On some invariants of Birkhoff billiards under conjugacy VL - 27 ER - TY - GEN AB - For any given positive integer l, we prove that every plane deformation of a circlewhich preserves the 1/2and 1/ (2l + 1) -rational caustics is trivial i.e. the deformationconsists only of similarities (rescalings and isometries). AU - Kaloshin, Vadim AU - Koudjinan, Edmond ID - 9435 TI - Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles ER - TY - BOOK AB - Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. In this groundbreaking book, Vadim Kaloshin and Ke Zhang provide the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. Kaloshin and Zhang follow Mather’s strategy but emphasize a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, this book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems. AU - Kaloshin, Vadim AU - Zhang, Ke ID - 8414 SN - 9-780-6912-0253-2 TI - Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom VL - 208 ER - TY - JOUR AB - We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit. AU - Bálint, Péter AU - De Simoi, Jacopo AU - Kaloshin, Vadim AU - Leguil, Martin ID - 8415 IS - 3 JF - Communications in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0010-3616 TI - Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards VL - 374 ER - TY - JOUR AB - For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set U in phase space of fixed measure, where the set of initial points which lead to collision is O(μ120) dense as μ→0. AU - Guardia, Marcel AU - Kaloshin, Vadim AU - Zhang, Jianlu ID - 8418 IS - 2 JF - Archive for Rational Mechanics and Analysis KW - Mechanical Engineering KW - Mathematics (miscellaneous) KW - Analysis SN - 0003-9527 TI - Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem VL - 233 ER - TY - JOUR AB - In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table Ω0 preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through Ω0. AU - Huang, Guan AU - Kaloshin, Vadim ID - 8416 IS - 2 JF - Moscow Mathematical Journal SN - 1609-4514 TI - On the finite dimensionality of integrable deformations of strictly convex integrable billiard tables VL - 19 ER - TY - JOUR AB - The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P+ (resp. P−). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity. On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion. AU - Delshams, Amadeu AU - Kaloshin, Vadim AU - de la Rosa, Abraham AU - Seara, Tere M. ID - 8417 IS - 3 JF - Communications in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0010-3616 TI - Global instability in the restricted planar elliptic three body problem VL - 366 ER - TY - JOUR AB - The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics. AU - Huang, Guan AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8422 IS - 2 JF - Geometric and Functional Analysis KW - Geometry and Topology KW - Analysis SN - 1016-443X TI - Nearly circular domains which are integrable close to the boundary are ellipses VL - 28 ER - TY - JOUR AB - The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains. AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8421 IS - 1 JF - Annals of Mathematics KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability SN - 0003-486X TI - On the local Birkhoff conjecture for convex billiards VL - 188 ER - TY - JOUR AB - In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’. AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8419 IS - 2131 JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences KW - General Engineering KW - General Physics and Astronomy KW - General Mathematics SN - 1364-503X TI - On the integrability of Birkhoff billiards VL - 376 ER - TY - JOUR AB - We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case. AU - Kaloshin, Vadim AU - Zhang, Ke ID - 8420 IS - 11 JF - Nonlinearity KW - Mathematical Physics KW - General Physics and Astronomy KW - Applied Mathematics KW - Statistical and Nonlinear Physics SN - 0951-7715 TI - Density of convex billiards with rational caustics VL - 31 ER - TY - JOUR AB - For any strictly convex planar domain Ω ⊂ R2 with a C∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and Ω¯ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sn}n≥1 (resp. {S¯n}n⩾1) of period going to infinity such that Sn and S¯n have the same period and perimeter for each n. AU - Buhovsky, Lev AU - Kaloshin, Vadim ID - 8426 JF - Regular and Chaotic Dynamics SN - 1560-3547 TI - Nonisometric domains with the same Marvizi-Melrose invariants VL - 23 ER - TY - JOUR AB - In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain. AU - Huang, Guan AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8423 IS - 1 JF - Duke Mathematical Journal SN - 0012-7094 TI - On the marked length spectrum of generic strictly convex billiard tables VL - 167 ER - TY - JOUR AB - We show that any sufficiently (finitely) smooth ℤ₂-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak. AU - De Simoi, Jacopo AU - Kaloshin, Vadim AU - Wei, Qiaoling ID - 8427 IS - 1 JF - Annals of Mathematics SN - 0003-486X TI - Dynamical spectral rigidity among Z2-symmetric strictly convex domains close to a circle VL - 186 ER - TY - JOUR AB - We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun–Jupiter–asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio μ=10−3 and a small eccentricity e0>0. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is ∼−ln(μe0)μ3/2e0. We expect our instability mechanism to apply to realistic values of e0 and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical ϵ-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol’d diffusion on time scales −lnϵϵ2. AU - Féjoz, Jacques AU - Guàrdia, Marcel AU - Kaloshin, Vadim AU - Roldán, Pablo ID - 8497 IS - 10 JF - Journal of the European Mathematical Society SN - 1435-9855 TI - Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem VL - 18 ER - TY - JOUR AU - Avila, Artur AU - De Simoi, Jacopo AU - Kaloshin, Vadim ID - 8496 IS - 2 JF - Annals of Mathematics SN - 0003-486X TI - An integrable deformation of an ellipse of small eccentricity is an ellipse VL - 184 ER - TY - JOUR AB - In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43, 1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior. AU - Guardia, M. AU - Kaloshin, Vadim AU - Zhang, J. ID - 8493 JF - Communications in Mathematical Physics SN - 0010-3616 TI - A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems VL - 348 ER - TY - JOUR AB - We prove a form of Arnold diffusion in the a-priori stable case. Let H0(p)+ϵH1(θ,p,t),θ∈Tn,p∈Bn,t∈T=R/T, be a nearly integrable system of arbitrary degrees of freedom n⩾2 with a strictly convex H0. We show that for a “generic” ϵH1, there exists an orbit (θ,p) satisfying ∥p(t)−p(0)∥>l(H1)>0, where l(H1) is independent of ϵ. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case. AU - Bernard, Patrick AU - Kaloshin, Vadim AU - Zhang, Ke ID - 8494 IS - 1 JF - Acta Mathematica SN - 0001-5962 TI - Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders VL - 217 ER -