@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},
}

@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},
}