Some remarks on the optimality of the Bruno-Rüssmann condition

Bounemoura, Abed

Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 2, pp. 341-353

We prove that the Bruno-Rüssmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz’s corresponding result for analytic circle diffeomorphisms and the uniqueness of invariant curves with a given irrational rotation number. We also prove a similar result for analytic Tonelli Hamiltonian flow with n=2 degrees of freedom; for n≥3 we only obtain a weaker result which recovers and slightly improves a theorem of Bessi.

Nous montrons que la condition de Bruno-Rüssmann est optimale pour la persistance de courbe invariante quasi-périodique analytique par une application twist analytique. La preuve repose sur le résultat analogue de Yoccoz pour un difféomorphisme analytique du cercle et sur l’unicité des courbes invariantes de nombre de rotation irrationnel. Nous montrons également un résultat similaire pour les Hamiltoniens Tonelli à n=2 degrés de liberté ; pour n≥3, nous obtenons un résultat plus faible qui généralise légèrement un théorème de Bessi.

Classification : 37J25, 37J40
Keywords: KAM theory, Twist maps, Tonelli Hamiltonians
Mots-clés : Théorie KAM

Bessi, U. An analytic counterexample to the KAM theorem, Ergodic Theory Dynam. Systems, Volume 20 (2000), pp. 317-333 | MR | Zbl | DOI

Brjuno, A.D. Analytical Form of Differential Equations I, Trans. Moscow Math. Soc., Volume 25 (1971), pp. 131-288 | Zbl | MR

Brjuno, A.D. Analytical Form of Differential Equations II, Trans. Moscow Math. Soc., Volume 26 (1972), pp. 199-239 | Zbl | MR

Bruno, A.D. Local Methods in Nonlinear Differential Equations, Springer, 1989 | MR | Zbl | DOI

Davie, A. M. The critical function for the semistandard map, Nonlinearity, Volume 7 (1994), pp. 219-229 | MR | Zbl | DOI

Fayad, B. R. Weak mixing for reparameterized linear flows on the torus, Ergodic Theory Dynam. Systems, Volume 22 (2002), pp. 187-201 | MR | Zbl | DOI

Fathi, A.; Giuliani, A; Sorrentino, A. Uniqueness of invariant Lagrangian graphs in a homology class or a cohomology class, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2009), pp. 659-680 | MR | Zbl | Numdam

Forni, G. Analytic destruction of invariant circles, Ergodic Theory Dynam. Systems, Volume 14 (1994), pp. 267-298 | MR | Zbl | DOI

Gentile, Guido Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers, Nonlinearity, Volume 28 (2015), pp. 2555-2585 | MR | Zbl | DOI

Geyer, L. Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 3661-3683 | MR | Zbl | DOI

Golé, Christophe Symplectic twist maps, Advanced Series in Nonlinear Dynamics, 18, World Scientific Publishing Co., Inc., 2001, xviii+305 pages (Global variational techniques) | MR | Zbl | DOI

Herman, M.-R. Sur les courbes invariantes par les difféomorphismes de l’anneau, Volume 1, Asterisque, Volume 103–104 (1983) | MR | Zbl | Numdam

Kuksin, S.; Pöschel, J. On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on dynamical systems (1994), pp. 96-116 (Birkhäuser) | MR | Zbl | DOI

Marmi, S. Critical functions for complex analytic maps, J. Phys. A, Volume 23 (1990), pp. 3447-3474 | MR | Zbl | DOI

Mather, J.N. A criterion for the non existence of invariant circles, Publ. Math. IHES, Volume 63 (1986) | MR | Zbl | Numdam | DOI

Mather, J.N. Destruction of invariant circles, Erg. Th. Dyn. Sys., Volume 8 (1988) | MR | Zbl

Moussa, P.; Marmi, S. Diophantine conditions and real or complex Brjuno functions, Noise, oscillators and algebraic randomness (Chapelle des Bois, 1999) (Lecture Notes in Phys.), Volume 550, Springer, 2000, pp. 324-342 | MR | Zbl | DOI

Pérez-Marco, R. Fixed points and circle maps, Acta Math., Volume 179 (1997), pp. 243-294 | MR | Zbl | DOI

Rüssmann, H. Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., Volume 6 (2001), pp. 119-204 | MR | Zbl | DOI

Rüssmann, H. On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, 1975, p. 598-624. Lecture Notes in Phys., Vol. 38 | MR | Zbl | DOI

Rüssmann, H. On the one-dimensional Schrodinger equation with a quasi-periodic potential, Ann. NY Acad. Sci., Volume 357 (1980), pp. 91-107 | MR | Zbl | DOI

Rüssmann, H. Non-degeneracy in the perturbation theory of integrable dynamical systems, Number theory and dynamical systems, Lond. Math. Soc. Lect. Note Ser. 134, 5–18, 1989 | MR | Zbl | DOI

Rüssmann, H. On the Frequencies of Quasi Periodic Solutions of Analytic Nearly Integrable Hamiltonian Systems, Seminar on Dynamical Systems (Kuksin, S.; Lazutkin, V.; Pöschel, J., eds.) (Progress in Nonlinear Differential Equations and Their Applications), Volume 12, Birkhäuser, 1994, pp. 160-183 | MR | Zbl | DOI

Salamon, D.A. The Kolmogorov-Arnold-Moser Theorem, Mathematical Physics Electronic Journal, Volume 10 (2004), pp. 1-37 | MR | Zbl

Sorrentino, A. Action-minimizing methods in Hamiltonian dynamics, Mathematical Notes, 50, Princeton University Press, Princeton, NJ, 2015, xii+115 pages (An introduction to Aubry-Mather theory) | MR

Treschev, D.; Zubelevich, O. Introduction to the perturbation theory of Hamiltonian systems, Springer Monographs in Mathematics, Springer, 2010, x+211 pages | MR | Zbl | DOI

Yoccoz, J.-C. Linéarisation des germes de difféomorphismes holomorphes de (C,0) , C. R. Acad. Sci. Paris Sér. I Math., Volume 306 (1988), pp. 55-58 | MR | Zbl

Yoccoz, J.-C. Small divisors in dimension one (Petits diviseurs en dimension 1), Astérisque. 231. Société Math. de France, 242 p., 1995 | Zbl | Numdam | MR