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

Topological conjugacies and behavior at infinity

Luis Barreira 1, and  Claudia Valls 2,

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.
Keywords: topological conjugacies., Exponential contractions.
Mathematics Subject Classification: Primary: 37D10, 37D2.
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), 58-90.  Google Scholar

[2]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241.  Google Scholar

[3]

G. Belickiĭ, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Functional Anal. Appl., 7 (1973), 268-277.  Google Scholar

[4]

G. Belickiĭ, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 107-177.  Google Scholar

[5]

D. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.  Google Scholar

[6]

D. Grobman, Topological classification of neighborhoods of a singularity in $n$-space, Mat. Sb. (N.S.), 56 (1962), 77-94.  Google Scholar

[7]

P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  Google Scholar

[8]

P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.  Google Scholar

[9]

P. McSwiggen, A geometric characterization of smooth linearizability, Michigan Math. J., 43 (1996), 321-335.  Google Scholar

[10]

J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ci., 40 (1968), 263-266.  Google Scholar

[11]

K. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  Google Scholar

[12]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.  Google Scholar

[13]

G. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  Google Scholar

[14]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.  Google Scholar

[15]

S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II., Amer. J. Math., 80 (1958), 623-631.  Google Scholar

show all references

References:
[1]

L. Barreira and C. Valls, Stable manifolds for nonautonomous equations without exponential dichotomy, J. Differential Equations, 221 (2006), 58-90.  Google Scholar

[2]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241.  Google Scholar

[3]

G. Belickiĭ, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Functional Anal. Appl., 7 (1973), 268-277.  Google Scholar

[4]

G. Belickiĭ, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 107-177.  Google Scholar

[5]

D. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.  Google Scholar

[6]

D. Grobman, Topological classification of neighborhoods of a singularity in $n$-space, Mat. Sb. (N.S.), 56 (1962), 77-94.  Google Scholar

[7]

P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  Google Scholar

[8]

P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.  Google Scholar

[9]

P. McSwiggen, A geometric characterization of smooth linearizability, Michigan Math. J., 43 (1996), 321-335.  Google Scholar

[10]

J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ci., 40 (1968), 263-266.  Google Scholar

[11]

K. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  Google Scholar

[12]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.  Google Scholar

[13]

G. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  Google Scholar

[14]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.  Google Scholar

[15]

S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II., Amer. J. Math., 80 (1958), 623-631.  Google Scholar

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