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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 and 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), 13; Russ. Math. Surv., 18 (1963), 9. [2] 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. [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), 159-176. [4] A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656. [5] A. D. Bruno, Analytical form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239. [6] X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312. [7] T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$, Bull. Soc. Math. France, 128 (2000), 69-85. [8] A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229. [9] 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). [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 "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. [12] W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen," VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. [13] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Oxford University Press, New York, 1979. [14] 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. [15] A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964. [16] S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen., 23 (1990), 3447-3474. [17] S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761. [18] 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. [19] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293. [20] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math.-Phys. Kl., (1962), 1-20. [21] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. [22] R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. [23] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422 and 8 (1882), 251-296. [24] H. Rüssmann, Über die iteration analytischer Funktionen, J. Math. Mech., 17 (1967), 523-532. [25] E. Schröder, Über iterierte Funktionen, Math. Ann., 3 (1871), 296-322. [26] C. L. Siegel, Iteration of analytic functions, Annals of Math., 43 (1942), 607-612. [27] 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. [28] C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin-Heidelberg-New York 1971. [29] J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88. [30] J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 125-173.

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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), 13; Russ. Math. Surv., 18 (1963), 9. [2] 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. [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), 159-176. [4] A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656. [5] A. D. Bruno, Analytical form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239. [6] X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312. [7] T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$, Bull. Soc. Math. France, 128 (2000), 69-85. [8] A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229. [9] 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). [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 "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. [12] W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen," VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. [13] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Oxford University Press, New York, 1979. [14] 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. [15] A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964. [16] S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen., 23 (1990), 3447-3474. [17] S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761. [18] 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. [19] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293. [20] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math.-Phys. Kl., (1962), 1-20. [21] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. [22] R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. [23] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422 and 8 (1882), 251-296. [24] H. Rüssmann, Über die iteration analytischer Funktionen, J. Math. Mech., 17 (1967), 523-532. [25] E. Schröder, Über iterierte Funktionen, Math. Ann., 3 (1871), 296-322. [26] C. L. Siegel, Iteration of analytic functions, Annals of Math., 43 (1942), 607-612. [27] 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. [28] C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin-Heidelberg-New York 1971. [29] J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88. [30] J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 125-173.
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