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), 58-90.  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

[1]

Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161

[2]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277

[3]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161

[4]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014

[5]

Alexander Gorodnik, Theron Hitchman, Ralf Spatzier. Regularity of conjugacies of algebraic actions of Zariski-dense groups. Journal of Modern Dynamics, 2008, 2 (3) : 509-540. doi: 10.3934/jmd.2008.2.509

[6]

Erdal Karapınar, Abdon Atangana, Andreea Fulga. Pata type contractions involving rational expressions with an application to integral equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3629-3640. doi: 10.3934/dcdss.2020420

[7]

Chris Good, Sergio Macías. What is topological about topological dynamics?. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1007-1031. doi: 10.3934/dcds.2018043

[8]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

[9]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[10]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[11]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555

[12]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

[13]

Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179

[14]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363

[15]

Christopher Oballe, Alan Cherne, Dave Boothe, Scott Kerick, Piotr J. Franaszczuk, Vasileios Maroulas. Bayesian topological signal processing. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021084

[16]

Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933

[17]

David Jerison, Nikola Kamburov. Free boundaries subject to topological constraints. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7213-7248. doi: 10.3934/dcds.2019301

[18]

John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293

[19]

John Banks. Topological mapping properties defined by digraphs. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83

[20]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]