@article{18483, abstract = {In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) by proving that a notion of rational integrability is equivalent to the C0-integrability condition used in their paper.}, author = {Kaloshin, Vadim and Koudjinan, Edmond and Zhang, Ke}, issn = {1420-8970}, journal = {Geometric and Functional Analysis}, pages = {1973--2007}, publisher = {Springer Nature}, title = {{Birkhoff conjecture for nearly centrally symmetric domains}}, doi = {10.1007/s00039-024-00695-6}, volume = {34}, year = {2024}, } @article{14427, abstract = {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.}, author = {Chen, Jianyu and Kaloshin, Vadim and Zhang, Hong Kun}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1--50}, publisher = {Springer Nature}, title = {{Length spectrum rigidity for piecewise analytic Bunimovich billiards}}, doi = {10.1007/s00220-023-04837-z}, volume = {404}, year = {2023}, } @article{12877, abstract = {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.}, author = {De Simoi, Jacopo and Kaloshin, Vadim and Leguil, Martin}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {829--901}, publisher = {Springer Nature}, title = {{Marked Length Spectral determination of analytic chaotic billiards with axial symmetries}}, doi = {10.1007/s00222-023-01191-8}, volume = {233}, year = {2023}, } @article{18959, abstract = {This workshop continues a series of workshops whose current format originated in 1981 under then-organizers Moser and Zehnder, and whose latest iteration took place in July 2023. The general goal of this series of workshops is to discuss the latest developments in the field of dynamical systems, broadly construed, and its connections with neighboring areas of mathematics such as differential geometry, partial differential equations, and more recently contact and symplectic geometry. We continued this tradition, bringing in new participants working in areas of dynamical systems and its connections with other areas of mathematics that are currently highly active and/or showing great promise for future development. Key focus areas for the 2023 workshop include spectral rigidity for planar domains, chaotic and oscillatory motions in celestial mechanics, conformal symplectic dynamics, and relations between dynamics.he workshop by the grant DMS-2230648, “US Junior Oberwolfach Fellows”.}, author = {Arnaud, Marie-Claude and Hutchings, Michael and Kaloshin, Vadim}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {3}, pages = {1671--1730}, publisher = {EMS Press}, title = {{Dynamische Systeme}}, doi = {10.4171/owr/2023/30}, volume = {20}, year = {2023}, } @article{12145, abstract = {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.}, author = {Koudjinan, Edmond and Kaloshin, Vadim}, issn = {1468-4845}, journal = {Regular and Chaotic Dynamics}, keywords = {Mechanical Engineering, Applied Mathematics, Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics, Mathematics (miscellaneous)}, number = {6}, pages = {525--537}, publisher = {Springer Nature}, title = {{On some invariants of Birkhoff billiards under conjugacy}}, doi = {10.1134/S1560354722050021}, volume = {27}, year = {2022}, } @article{17063, abstract = {This workshop continued a biannual series of workshops at Oberwolfach on dynamical systems that started with a meeting organized by Moser and Zehnder in 1981. Workshops in this series focus on new results and developments in dynamical systems and related areas of mathematics, with symplectic geometry playing an important role in recent years in connection with Hamiltonian dynamics. In this year special emphasis was placed on various kinds of spectra (in contact geometry, in Riemannian geometry, in dynamical systems and in symplectic topology) and their applications to dynamics.}, author = {Arnaud, Marie-Claude and Hofer, Helmut W. and Hutchings, Michael and Kaloshin, Vadim}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {3}, pages = {1735--1803}, publisher = {European Mathematical Society}, title = {{Dynamische Systeme}}, doi = {10.4171/owr/2021/33}, volume = {18}, year = {2022}, } @unpublished{9435, abstract = {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).}, author = {Kaloshin, Vadim and Koudjinan, Edmond}, booktitle = {arXiv}, title = {{Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles}}, doi = {10.48550/arXiv.2107.03499}, year = {2021}, } @book{8414, abstract = {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.}, author = {Kaloshin, Vadim and Zhang, Ke}, isbn = {9-780-6912-0253-2}, pages = {224}, publisher = {Princeton University Press}, title = {{Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom}}, doi = {10.1515/9780691204932}, volume = {208}, year = {2020}, } @article{8415, abstract = {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.}, author = {Bálint, Péter and De Simoi, Jacopo and Kaloshin, Vadim and Leguil, Martin}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {3}, pages = {1531--1575}, publisher = {Springer Nature}, title = {{Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards}}, doi = {10.1007/s00220-019-03448-x}, volume = {374}, year = {2019}, } @article{8416, abstract = {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.}, author = {Huang, Guan and Kaloshin, Vadim}, issn = {1609-4514}, journal = {Moscow Mathematical Journal}, number = {2}, pages = {307--327}, publisher = {American Mathematical Society}, title = {{On the finite dimensionality of integrable deformations of strictly convex integrable billiard tables}}, doi = {10.17323/1609-4514-2019-19-2-307-327}, volume = {19}, year = {2019}, } @article{8418, abstract = {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.}, author = {Guardia, Marcel and Kaloshin, Vadim and Zhang, Jianlu}, issn = {0003-9527}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis}, number = {2}, pages = {799--836}, publisher = {Springer Nature}, title = {{Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem}}, doi = {10.1007/s00205-019-01368-7}, volume = {233}, year = {2019}, } @article{8417, abstract = {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.}, author = {Delshams, Amadeu and Kaloshin, Vadim and de la Rosa, Abraham and Seara, Tere M.}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {3}, pages = {1173--1228}, publisher = {Springer Nature}, title = {{Global instability in the restricted planar elliptic three body problem}}, doi = {10.1007/s00220-018-3248-z}, volume = {366}, year = {2018}, } @article{8419, abstract = {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’.}, author = {Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {1364-503X}, journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, keywords = {General Engineering, General Physics and Astronomy, General Mathematics}, number = {2131}, publisher = {The Royal Society}, title = {{On the integrability of Birkhoff billiards}}, doi = {10.1098/rsta.2017.0419}, volume = {376}, year = {2018}, } @article{8420, abstract = {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.}, author = {Kaloshin, Vadim and Zhang, Ke}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {11}, pages = {5214--5234}, publisher = {IOP Publishing}, title = {{Density of convex billiards with rational caustics}}, doi = {10.1088/1361-6544/aadc12}, volume = {31}, year = {2018}, } @article{8421, abstract = {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.}, author = {Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {0003-486X}, journal = {Annals of Mathematics}, keywords = {Statistics, Probability and Uncertainty, Statistics and Probability}, number = {1}, pages = {315--380}, publisher = {Annals of Mathematics, Princeton U}, title = {{On the local Birkhoff conjecture for convex billiards}}, doi = {10.4007/annals.2018.188.1.6}, volume = {188}, year = {2018}, } @article{8422, abstract = {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.}, author = {Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {1016-443X}, journal = {Geometric and Functional Analysis}, keywords = {Geometry and Topology, Analysis}, number = {2}, pages = {334--392}, publisher = {Springer Nature}, title = {{Nearly circular domains which are integrable close to the boundary are ellipses}}, doi = {10.1007/s00039-018-0440-4}, volume = {28}, year = {2018}, } @article{8426, abstract = {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.}, author = {Buhovsky, Lev and Kaloshin, Vadim}, issn = {1560-3547}, journal = {Regular and Chaotic Dynamics}, pages = {54--59}, publisher = {Springer Nature}, title = {{Nonisometric domains with the same Marvizi-Melrose invariants}}, doi = {10.1134/s1560354718010057}, volume = {23}, year = {2018}, } @article{8423, abstract = {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.}, author = {Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {0012-7094}, journal = {Duke Mathematical Journal}, number = {1}, pages = {175--209}, publisher = {Duke University Press}, title = {{On the marked length spectrum of generic strictly convex billiard tables}}, doi = {10.1215/00127094-2017-0038}, volume = {167}, year = {2017}, } @article{8427, abstract = {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.}, author = {De Simoi, Jacopo and Kaloshin, Vadim and Wei, Qiaoling}, issn = {0003-486X}, journal = {Annals of Mathematics}, number = {1}, pages = {277--314}, publisher = {Annals of Mathematics}, title = {{Dynamical spectral rigidity among Z2-symmetric strictly convex domains close to a circle}}, doi = {10.4007/annals.2017.186.1.7}, volume = {186}, year = {2017}, } @article{8493, abstract = {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.}, author = {Guardia, M. and Kaloshin, Vadim and Zhang, J.}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, pages = {321--361}, publisher = {Springer Nature}, title = {{A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems}}, doi = {10.1007/s00220-016-2705-9}, volume = {348}, year = {2016}, }