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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:
 [1] 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). [2] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Usp. Math. Nauk {\bf 18} (1963), 18 (1963). doi: 10.1070/RM1963v018n06ABEH001143. [3] 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. [4] A. Berretti and G. Gentile, Bryuno function and the standard map,, Comm. Math. Phys., 220 (2001), 623. [5] A. D. Bruno, Analytical form of differential equations,, Trans. Moscow Math. Soc., 25 (1971), 131. [6] X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks,, Ann. of Math., 164 (2006), 265. [7] T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$,, Bull. Soc. Math. France, 128 (2000), 69. [8] A. M. Davie, The critical function for the semistandard map,, Nonlinearity, 7 (1994), 219. [9] A. Giorgilli, Quantitative methods in classical perturbation theory,, in Proceedings of the NATO ASI school, 336 (1995), 21. [10] A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions,, MPEJ, 3 (1997). [11] A. Giorgilli and U. Locatelli, A classical self-contained proof of Kolmogorov's theorem on invariant tori,, in Proceedings of the NATO ASI school, 533 (1999), 72. [12] W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen,", VEB Deutscher Verlag der Wissenschaften, (1967). [13] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Oxford University Press, (1979). [14] A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 71. [15] A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964). [16] S. Marmi, Critical functions for complex analytic maps,, J. Phys. A: Math. Gen., 23 (1990), 3447. [17] S. Marmi and J. Stark, On the standard map critical function,, Nonlinearity, 5 (1992), 743. [18] S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors,, in, 1784 (2002), 175. [19] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties,, Comm. Math. Phys., 186 (1997), 265. [20] J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Gött., (1962), 1. [21] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. [22] R. Pérez-Marco, Fixed points and circle maps,, Acta Math., 179 (1997), 243. [23] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle,, Journal de Mathématiques, 7 (1881), 375. [24] H. Rüssmann, Über die iteration analytischer Funktionen,, J. Math. Mech., 17 (1967), 523. [25] E. Schröder, Über iterierte Funktionen,, Math. Ann., 3 (1871), 296. [26] C. L. Siegel, Iteration of analytic functions,, Annals of Math., 43 (1942), 607. [27] C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung,, Nachr. Akad. Wiss. Göttingen, (1952), 21. [28] C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). [29] J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3. [30] J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms,, in, 1784 (2002), 125.

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##### References:
 [1] 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). [2] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Usp. Math. Nauk {\bf 18} (1963), 18 (1963). doi: 10.1070/RM1963v018n06ABEH001143. [3] 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. [4] A. Berretti and G. Gentile, Bryuno function and the standard map,, Comm. Math. Phys., 220 (2001), 623. [5] A. D. Bruno, Analytical form of differential equations,, Trans. Moscow Math. Soc., 25 (1971), 131. [6] X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks,, Ann. of Math., 164 (2006), 265. [7] T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$,, Bull. Soc. Math. France, 128 (2000), 69. [8] A. M. Davie, The critical function for the semistandard map,, Nonlinearity, 7 (1994), 219. [9] A. Giorgilli, Quantitative methods in classical perturbation theory,, in Proceedings of the NATO ASI school, 336 (1995), 21. [10] A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions,, MPEJ, 3 (1997). [11] A. Giorgilli and U. Locatelli, A classical self-contained proof of Kolmogorov's theorem on invariant tori,, in Proceedings of the NATO ASI school, 533 (1999), 72. [12] W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen,", VEB Deutscher Verlag der Wissenschaften, (1967). [13] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Oxford University Press, (1979). [14] A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 71. [15] A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964). [16] S. Marmi, Critical functions for complex analytic maps,, J. Phys. A: Math. Gen., 23 (1990), 3447. [17] S. Marmi and J. Stark, On the standard map critical function,, Nonlinearity, 5 (1992), 743. [18] S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors,, in, 1784 (2002), 175. [19] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties,, Comm. Math. Phys., 186 (1997), 265. [20] J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Gött., (1962), 1. [21] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. [22] R. Pérez-Marco, Fixed points and circle maps,, Acta Math., 179 (1997), 243. [23] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle,, Journal de Mathématiques, 7 (1881), 375. [24] H. Rüssmann, Über die iteration analytischer Funktionen,, J. Math. Mech., 17 (1967), 523. [25] E. Schröder, Über iterierte Funktionen,, Math. Ann., 3 (1871), 296. [26] C. L. Siegel, Iteration of analytic functions,, Annals of Math., 43 (1942), 607. [27] C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung,, Nachr. Akad. Wiss. Göttingen, (1952), 21. [28] C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). [29] J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3. [30] J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms,, in, 1784 (2002), 125.
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