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December  2010, 3(4): 601-621. doi: 10.3934/dcdss.2010.3.601

Convergence radius in the Poincaré-Siegel problem

 1 Dipartimento di Matematica, Universitµa degli Studi di Milano, via Saldini 50, 20133 | Milano, Italy 2 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa (PI)

Received  March 2009 Revised  May 2010 Published  August 2010

We reconsider the Poincaré-Siegel center problem, namely the problem of conjugating an analytic system of differential equations in the neighbourhood of an equilibrium to its linear part $\Lambda=\diag(\lambda_1,\ldots,\lambda_n)$. If the linear part is non--resonant we show that the convergence radius $r$ of the conjugating transformation satisfies $\log r(\Lambda )\ge -CB+C'$ with $C=1$ and a constant $C'$ not depending on $\Lambda$. The convergence condition is the same as the Bruno condition since $B = -\sum_{r\ge 1}2^{-r}\log\alpha_{2^r-1}$, where $\alpha_r = \min_{0\le s\le r} \beta_r$ for $r\ge 0$ and $\beta_r = \min_{j=1,\ldots,n}\ \ \ \min_{k\inZ_+^n,|k|=r+1}$ | < k, $\lambda$ > $- \lambda_j$|. Our lower bound improves the previous results for $n\gt 1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for the discrete time version of the center problem when $n=1$, namely the linearization problem for germs of holomorphic maps when the eigenvalue of the fixed point is on the unit circle.
Citation: Antonio Giorgilli, Stefano Marmi. Convergence radius in the Poincaré-Siegel problem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 601-621. doi: 10.3934/dcdss.2010.3.601
References:
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show all references

References:
  V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian, Usp. Mat. Nauk, 18 (1963), 13; Russ. Math. Surv., 18 (1963), 9. Google Scholar  V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Usp. Math. Nauk 18 (1963), 91; Russ. Math. Surv., 18 (1963), 85. doi: 10.1070/RM1963v018n06ABEH001143. Google Scholar  A. Berretti and G. Gentile, Scaling properties for the radius of convergence of a Lindstedt series: the standard map, J. Math. Pures Appl., 78 (1999), 159-176. Google Scholar  A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656. Google Scholar  A. D. Bruno, Analytical form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239. Google Scholar  X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312. Google Scholar  T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$, Bull. Soc. Math. France, 128 (2000), 69-85. Google Scholar  A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229. Google Scholar  A. Giorgilli, Quantitative methods in classical perturbation theory, in Proceedings of the NATO ASI school "From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in $N$-body Dynamical Systems,'' A.E. Roy and B.A. Steves eds., NATO ASI Series B: Physics, Vol. 336, Plenum Press, New York, 21-37 (1995). Google Scholar  A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions, MPEJ, 3 (1997). Google Scholar  A. Giorgilli and U. Locatelli, A classical self-contained proof of Kolmogorov's theorem on invariant tori, in Proceedings of the NATO ASI school "Hamiltonian systems with three or more degrees of freedom,'' C. Simó ed., NATO ASI series C: Math. Phys. Sci., Vol. 533, Kluwer Academic Publishers, Dordrecht-Boston-London, (1999), 72-89. Google Scholar  W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen," VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. Google Scholar  G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Oxford University Press, New York, 1979. Google Scholar  A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, Dokl. Akad. Nauk SSSR, 98 (1954), 527; English translation in: Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in: G. Casati and J. Ford: Stochastic behavior in classical and quantum Hamiltonian systems, Lecture Notes in Physics, 93 (1979), 51-56. Google Scholar  A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964. Google Scholar  S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen., 23 (1990), 3447-3474. Google Scholar  S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761. Google Scholar  S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 175-191. Google Scholar  S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293. Google Scholar  J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math.-Phys. Kl., (1962), 1-20. Google Scholar  J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. Google Scholar  R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. Google Scholar  H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422 and 8 (1882), 251-296. Google Scholar  H. Rüssmann, Über die iteration analytischer Funktionen, J. Math. Mech., 17 (1967), 523-532. Google Scholar  E. Schröder, Über iterierte Funktionen, Math. Ann., 3 (1871), 296-322. Google Scholar  C. L. Siegel, Iteration of analytic functions, Annals of Math., 43 (1942), 607-612. Google Scholar  C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. Math.-Phys.-Chem. Abt., (1952), 21-30. Google Scholar  C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin-Heidelberg-New York 1971. Google Scholar  J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88. Google Scholar  J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 125-173. Google Scholar
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