American Institute of Mathematical Sciences

December  2012, 32(12): 4321-4360. doi: 10.3934/dcds.2012.32.4321

Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps

 1 School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30332-0160, United States 2 Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, United States

Received  June 2011 Revised  March 2012 Published  August 2012

We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.
Citation: Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
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References:
 [1] Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229 [2] Simeon Reich, Alexander J. Zaslavski. Convergence of generic infinite products of homogeneous order-preserving mappings. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 929-945. doi: 10.3934/dcds.1999.5.929 [3] Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006 [4] Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179 [5] Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425 [6] Fengbo Hang, Fanghua Lin. Topology of Sobolev mappings IV. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1097-1124. doi: 10.3934/dcds.2005.13.1097 [7] Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451 [8] B. S. Lee, Arif Rafiq. Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 287-293. doi: 10.3934/naco.2014.4.287 [9] Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385 [10] Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557 [11] Jialin Hong, Lijun Miao, Liying Zhang. Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4295-4315. doi: 10.3934/dcdsb.2019120 [12] Dariusz Bugajewski, Piotr Kasprzak. On mappings of higher order and their applications to nonlinear equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 627-647. doi: 10.3934/cpaa.2012.11.627 [13] Alexei Pokrovskii, Oleg Rasskazov. Structure of index sequences for mappings with an asymptotic derivative. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 653-670. doi: 10.3934/dcds.2007.17.653 [14] Tomas Persson. Typical points and families of expanding interval mappings. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4019-4034. doi: 10.3934/dcds.2017170 [15] Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631 [16] Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2018138 [17] Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018 [18] Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57 [19] H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361 [20] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205

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