@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{8689, abstract = {This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the Whitney conjugacy of a non{degenerate, real{analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov's set are provided in the case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the d-torus and (B) a domain with C2 boundary times the d-torus. All constants are explicitly given.}, author = {Chierchia, Luigi and Koudjinan, Edmond}, issn = {1560-3547}, journal = {Regular and Chaotic Dynamics}, keywords = {Nearly{integrable Hamiltonian systems, perturbation theory, KAM Theory, Arnold's scheme, Kolmogorov's set, primary invariant tori, Lagrangian tori, measure estimates, small divisors, integrability on nowhere dense sets, Diophantine frequencies.}, number = {1}, pages = {61--88}, publisher = {Springer Nature}, title = {{V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates}}, doi = {10.1134/S1560354721010044}, volume = {26}, year = {2021}, } @unpublished{9435, abstract = {For any given positive integer l, we prove that every plane deformation of a circlewhich preserves the 1/2and 1/ (2l + 1) -rational caustics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}, title = {{Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles}}, year = {2021}, } @article{8691, abstract = {Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.}, author = {Koudjinan, Edmond}, issn = {0022-0396}, journal = {Journal of Differential Equations}, keywords = {Analysis}, number = {6}, pages = {4720--4750}, publisher = {Elsevier}, title = {{A KAM theorem for finitely differentiable Hamiltonian systems}}, doi = {10.1016/j.jde.2020.03.044}, volume = {269}, year = {2020}, } @article{8694, abstract = {We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps fa(x)=a−x2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition 𝒫 of the parameter interval Ω=[1.4,2] into almost 4×106 subintervals, for each of which we compute to high precision the orbits of the critical points up to some time N and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide 𝒫 into a family 𝒫+ of intervals, which we call stochastic intervals, and a family 𝒫− of intervals, which we call regular intervals. We numerically prove that each interval ω∈𝒫+ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in ω has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in 𝒫+ are stochastic and most parameters belonging to the intervals in 𝒫− are regular, thus the names. We prove that the intervals in 𝒫+ occupy almost 90% of the total measure of Ω. The software and the data are freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parameterized families of dynamical systems.}, author = {Golmakani, Ali and Koudjinan, Edmond and Luzzatto, Stefano and Pilarczyk, Pawel}, journal = {Chaos}, number = {7}, publisher = {AIP}, title = {{Rigorous numerics for critical orbits in the quadratic family}}, doi = {10.1063/5.0012822}, volume = {30}, year = {2020}, } @article{8693, abstract = {We review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.}, author = {Chierchia, Luigi and Koudjinan, Edmond}, journal = {Regular and Chaotic Dynamics}, pages = {583–606}, publisher = {Springer}, title = {{V. I. Arnold’s “pointwise” KAM theorem}}, doi = {10.1134/S1560354719060017}, volume = {24}, year = {2019}, }