July  2022, 42(7): 3077-3101. doi: 10.3934/dcds.2022009

Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum

1. 

IMJ-PRG, Université de Paris, Bâtiment Sophie Germain, Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13, France

2. 

Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

*Corresponding author: Claire Chavaudret

Received  August 2021 Published  July 2022 Early access  February 2022

Fund Project: The second author is supported by the Dynamics and Information Theory Institute at the Scuola Normale Superiore and the MIUR PRIN Project Regular and stochastic behaviour in dynamical systems nr. 2017S35EHN.

One considers a system on $ \mathbb{C}^2 $ close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of $ d\alpha $, where $ d\in \mathbb{N}^* $ and $ \alpha $ is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.

Citation: Claire Chavaudret, Stefano Marmi. Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3077-3101. doi: 10.3934/dcds.2022009
References:
[1]

J. Aurouet, Normalisation de Champs de Vecteurs Holomorphes et Équations Différentielles Implicites, Ph.D thesis, University of Nice, 2013

[2]

A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656.  doi: 10.1007/s002200100456.

[3]

A. Berretti and G. Gentile, Periodic and quasi-periodic orbits for the standard map, Comm. Math. Phys., 231 (2002), 135-156.  doi: 10.1007/s00220-002-0674-7.

[4]

A. BerrettiS. Marmi and D. Sauzin, Limit at resonances of linearizations of some complex analytic dynamical systems, Ergodic Theory Dynam. Systems, 20 (2000), 963-990.  doi: 10.1017/S0143385700000547.

[5]

A. D. Brjuno, Analytic forms of differential equations, Trudy Moskov. Mat. Obšč., 25 (1971), 119–262; 26 (1972), 199–239.

[6]

X. Buff and A. Chéritat, Upper bound for the size of quadratic Siegel disks, Invent. Math., 156 (2004), 1-24.  doi: 10.1007/s00222-003-0331-6.

[7]

X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312.  doi: 10.4007/annals.2006.164.265.

[8]

T. Carletti and S. Marmi, Linearization of analytic and non analytic germs of diffeomorphisms of $ (\mathbb{C},0)$, Bull. Soc. Math. France, 128 (2000), 69-85.  doi: 10.24033/bsmf.2363.

[9]

C. Carminati and S. Marmi, Linearization of germs: Regular dependence on the multiplier, Bull. Soc. Math. France, 136 (2008), 533-564.  doi: 10.24033/bsmf.2565.

[10]

C. Chavaudret, Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition, Ann. Inst. Fourier, 66 (2016), 1987-2020.  doi: 10.5802/aif.3055.

[11]

C. Chavaudret and S. Marmi, Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition, J. Mod. Dyn., 6 (2012), 59-78.  doi: 10.3934/jmd.2012.6.59.

[12]

D. Cheraghi and A. Chéritat, A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type, Invent. Math., 202 (2015), 677-742.  doi: 10.1007/s00222-014-0576-2.

[13]

A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229.  doi: 10.1088/0951-7715/7/1/009.

[14]

G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.  doi: 10.1007/s10955-006-9127-6.

[15]

A. Giorgilli and S. Marmi, Improved estimates for the convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S 3, 4 (2010), 601-621.  doi: 10.3934/dcdss.2010.3.601.

[16]

A. GiorgilliU. Locatelli and M. Sansottera, Improved convergence estimates for the Schröder-Siegel problem, Ann. Mat. Pura Appl., 194 (2015), 995-1013.  doi: 10.1007/s10231-014-0408-4.

[17]

J. M. Greene and I. C. Percival, Hamiltonian maps in the complex plane, Phys. D, 3 (1981), 530-548.  doi: 10.1016/0167-2789(81)90038-5.

[18]

J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.

[19]

S. Marmi, Critical functions for complex analytic maps, J. Phys. A, 23 (1990), 3447-3474.  doi: 10.1088/0305-4470/23/15/019.

[20]

S. MarmiP. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.  doi: 10.1007/s002200050110.

[21]

S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761.  doi: 10.1088/0951-7715/5/3/007.

[22]

L. Stolovitch, Singular complete integrability, Inst. Hautes Études Sci. Publ. Math., 91 (2001), 133-210. 

[23]

J-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995), 1-88. 

show all references

References:
[1]

J. Aurouet, Normalisation de Champs de Vecteurs Holomorphes et Équations Différentielles Implicites, Ph.D thesis, University of Nice, 2013

[2]

A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656.  doi: 10.1007/s002200100456.

[3]

A. Berretti and G. Gentile, Periodic and quasi-periodic orbits for the standard map, Comm. Math. Phys., 231 (2002), 135-156.  doi: 10.1007/s00220-002-0674-7.

[4]

A. BerrettiS. Marmi and D. Sauzin, Limit at resonances of linearizations of some complex analytic dynamical systems, Ergodic Theory Dynam. Systems, 20 (2000), 963-990.  doi: 10.1017/S0143385700000547.

[5]

A. D. Brjuno, Analytic forms of differential equations, Trudy Moskov. Mat. Obšč., 25 (1971), 119–262; 26 (1972), 199–239.

[6]

X. Buff and A. Chéritat, Upper bound for the size of quadratic Siegel disks, Invent. Math., 156 (2004), 1-24.  doi: 10.1007/s00222-003-0331-6.

[7]

X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312.  doi: 10.4007/annals.2006.164.265.

[8]

T. Carletti and S. Marmi, Linearization of analytic and non analytic germs of diffeomorphisms of $ (\mathbb{C},0)$, Bull. Soc. Math. France, 128 (2000), 69-85.  doi: 10.24033/bsmf.2363.

[9]

C. Carminati and S. Marmi, Linearization of germs: Regular dependence on the multiplier, Bull. Soc. Math. France, 136 (2008), 533-564.  doi: 10.24033/bsmf.2565.

[10]

C. Chavaudret, Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition, Ann. Inst. Fourier, 66 (2016), 1987-2020.  doi: 10.5802/aif.3055.

[11]

C. Chavaudret and S. Marmi, Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition, J. Mod. Dyn., 6 (2012), 59-78.  doi: 10.3934/jmd.2012.6.59.

[12]

D. Cheraghi and A. Chéritat, A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type, Invent. Math., 202 (2015), 677-742.  doi: 10.1007/s00222-014-0576-2.

[13]

A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229.  doi: 10.1088/0951-7715/7/1/009.

[14]

G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.  doi: 10.1007/s10955-006-9127-6.

[15]

A. Giorgilli and S. Marmi, Improved estimates for the convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S 3, 4 (2010), 601-621.  doi: 10.3934/dcdss.2010.3.601.

[16]

A. GiorgilliU. Locatelli and M. Sansottera, Improved convergence estimates for the Schröder-Siegel problem, Ann. Mat. Pura Appl., 194 (2015), 995-1013.  doi: 10.1007/s10231-014-0408-4.

[17]

J. M. Greene and I. C. Percival, Hamiltonian maps in the complex plane, Phys. D, 3 (1981), 530-548.  doi: 10.1016/0167-2789(81)90038-5.

[18]

J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.

[19]

S. Marmi, Critical functions for complex analytic maps, J. Phys. A, 23 (1990), 3447-3474.  doi: 10.1088/0305-4470/23/15/019.

[20]

S. MarmiP. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.  doi: 10.1007/s002200050110.

[21]

S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761.  doi: 10.1088/0951-7715/5/3/007.

[22]

L. Stolovitch, Singular complete integrability, Inst. Hautes Études Sci. Publ. Math., 91 (2001), 133-210. 

[23]

J-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995), 1-88. 

Figure 1.  The Brjuno function of $ N\alpha $ for $ \alpha \in (0, 1) $. Red: $ N = 1 $; green: $ N = 2 $; blue: $ N = 3 $; purple: $ N = 4 $
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