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Non co-preservation of the & -rational caustics along deformations of circles
V. Kaloshin, C. E. Koudjinan
For any given positive integer , we prove that every plane deformation of a circle which preserves the and -rational caustics is trivial i.e. the deformation con…
On some invariants of Birkhoff billiards under conjugacy
V. Kaloshin, C. E. Koudjinan
In the class of strictly convex smooth boundaries, each of which not having strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the "no…
V.I. Arnold's "Global" KAM Theorem and geometric measure estimates
L. Chierchia, C. E. Koudjinan
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 `gl…
Rigorous numerics for critical orbits in the quadratic family
Ali Golmakani, Comlan Edmond Koudjinan, Stefano Luzzatto +1
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 on…
V.I. Arnold's "pointwise" KAM Theorem
Luigi Chierchia, Comlan Edmond Koudjinan
We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in t…
A KAM Theorem for finitely differentiable Hamiltonian systems
Comlan Edmond Koudjinan
Given , we prove the persistence of a Cantor--family of KAM tori of measure for any non--degenerate nearly integrable Hamiltonian system o…