# American Institute of Mathematical Sciences

March  2014, 13(2): 687-701. doi: 10.3934/cpaa.2014.13.687

## Topological conjugacies and behavior at infinity

 1 Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received  March 2013 Revised  June 2013 Published  October 2013

We obtain a version of the Grobman--Hartman theorem in Banach spaces for perturbations of a nonuniform exponential contraction, both for discrete and continuous time. More precisely, we consider the general case of an exponential contraction with an arbitrary nonuniform part and obtained from a nonautonomous dynamics, and we establish the existence of Hölder continuous conjugacies between an exponential contraction and any sufficiently small perturbation. As a nontrivial application, we describe the asymptotic behavior of the topological conjugacies in terms of the perturbations: namely, we show that for perturbations growing in a certain controlled manner the conjugacies approach zero at infinity and that when the perturbations decay exponentially at infinity the conjugacies have the same exponential behavior.
Citation: Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure & Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687
##### References:
 [1] L. Barreira and C. Valls, Stable manifolds for nonautonomous equations without exponential dichotomy,, J. Differential Equations, 221 (2006), 58.   Google Scholar [2] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity,, J. Dynam. Differential Equations, 19 (2007), 215.   Google Scholar [3] G. Belickiĭ, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class,, Functional Anal. Appl., 7 (1973), 268.   Google Scholar [4] G. Belickiĭ, Equivalence and normal forms of germs of smooth mappings,, Russian Math. Surveys, 33 (1978), 107.   Google Scholar [5] D. Grobman, Homeomorphism of systems of differential equations,, Dokl. Akad. Nauk SSSR, 128 (1959), 880.   Google Scholar [6] D. Grobman, Topological classification of neighborhoods of a singularity in $n$-space,, Mat. Sb. (N.S.), 56 (1962), 77.   Google Scholar [7] P. Hartman, A lemma in the theory of structural stability of differential equations,, Proc. Amer. Math. Soc., 11 (1960), 610.   Google Scholar [8] P. Hartman, On the local linearization of differential equations,, Proc. Amer. Math. Soc., 14 (1963), 568.   Google Scholar [9] P. McSwiggen, A geometric characterization of smooth linearizability,, Michigan Math. J., 43 (1996), 321.   Google Scholar [10] J. Palis, On the local structure of hyperbolic points in Banach spaces,, An. Acad. Brasil. Ci., 40 (1968), 263.   Google Scholar [11] K. Palmer, A generalization of Hartman's linearization theorem,, J. Math. Anal. Appl., 41 (1973), 753.   Google Scholar [12] C. Pugh, On a theorem of P. Hartman,, Amer. J. Math., 91 (1969), 363.   Google Scholar [13] G. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035.   Google Scholar [14] S. Sternberg, Local contractions and a theorem of Poincaré,, Amer. J. Math., 79 (1957), 809.   Google Scholar [15] S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II.,, Amer. J. Math., 80 (1958), 623.   Google Scholar

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##### References:
 [1] L. Barreira and C. Valls, Stable manifolds for nonautonomous equations without exponential dichotomy,, J. Differential Equations, 221 (2006), 58.   Google Scholar [2] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity,, J. Dynam. Differential Equations, 19 (2007), 215.   Google Scholar [3] G. Belickiĭ, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class,, Functional Anal. Appl., 7 (1973), 268.   Google Scholar [4] G. Belickiĭ, Equivalence and normal forms of germs of smooth mappings,, Russian Math. Surveys, 33 (1978), 107.   Google Scholar [5] D. Grobman, Homeomorphism of systems of differential equations,, Dokl. Akad. Nauk SSSR, 128 (1959), 880.   Google Scholar [6] D. Grobman, Topological classification of neighborhoods of a singularity in $n$-space,, Mat. Sb. (N.S.), 56 (1962), 77.   Google Scholar [7] P. Hartman, A lemma in the theory of structural stability of differential equations,, Proc. Amer. Math. Soc., 11 (1960), 610.   Google Scholar [8] P. Hartman, On the local linearization of differential equations,, Proc. Amer. Math. Soc., 14 (1963), 568.   Google Scholar [9] P. McSwiggen, A geometric characterization of smooth linearizability,, Michigan Math. J., 43 (1996), 321.   Google Scholar [10] J. Palis, On the local structure of hyperbolic points in Banach spaces,, An. Acad. Brasil. Ci., 40 (1968), 263.   Google Scholar [11] K. Palmer, A generalization of Hartman's linearization theorem,, J. Math. Anal. Appl., 41 (1973), 753.   Google Scholar [12] C. Pugh, On a theorem of P. Hartman,, Amer. J. Math., 91 (1969), 363.   Google Scholar [13] G. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035.   Google Scholar [14] S. Sternberg, Local contractions and a theorem of Poincaré,, Amer. J. Math., 79 (1957), 809.   Google Scholar [15] S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II.,, Amer. J. Math., 80 (1958), 623.   Google Scholar
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