zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A Grobman--Hartman theorem for nonuniformly hyperbolic dynamics. (English) Zbl 1099.37022
Summary: We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps $A_{m}+f_{m}$ such that the linear parts $A_{m}$ admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps $A_{m}+f_{m}$ and $A_{m}$. Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations $f_{m}$: we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.

37D25Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37B55Nonautonomous dynamical systems
37C15Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
34D09Dichotomy, trichotomy
34D08Characteristic and Lyapunov exponents
Full Text: DOI
[1] Barreira, L.; Valls, C.: Stable manifolds for nonautonomous equations without exponential dichotomy. J. differential equations 221, 58-90 (2006) · Zbl 1098.34036
[2] L. Barreira, C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, preprint · Zbl 1123.34040
[3] Belickiĭ, G.: Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class. Funct. anal. Appl. 7, 268-277 (1973)
[4] Belickiĭ, G.: Equivalence and normal forms of germs of smooth mappings. Russian math. Surveys 33, 107-177 (1978) · Zbl 0398.58009
[5] Grobman, D.: Homeomorphism of systems of differential equations. Dokl. akad. Nauk SSSR 128, 880-881 (1959) · Zbl 0100.29804
[6] Grobman, D.: Topological classification of neighborhoods of a singularity in n-space. Mat. sb. (N.S.) 56, No. 98, 77-94 (1962) · Zbl 0142.34402
[7] Guysinsky, M.; Hasselblatt, B.; Rayskin, V.: Differentiability of the hartman -- grobman linearization. Discrete contin. Dyn. syst. 9, 979-984 (2003) · Zbl 1024.37022
[8] Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. amer. Math. soc. 11, 610-620 (1960) · Zbl 0132.31904
[9] Hartman, P.: On the local linearization of differential equations. Proc. amer. Math. soc. 14, 568-573 (1963) · Zbl 0115.29801
[10] Mcswiggen, P.: A geometric characterization of smooth linearizability. Michigan math. J. 43, 321-335 (1996) · Zbl 0868.34034
[11] Moser, J.: On a theorem of Anosov. J. differential equations 5, 411-440 (1969) · Zbl 0169.42303
[12] Palis, J.: On the local structure of hyperbolic points in Banach spaces. An. acad. Brasil. ciênc. 40, 263-266 (1968) · Zbl 0184.17803
[13] Palmer, K.: A generalization of hartman’s linearization theorem. J. math. Anal. appl. 41, 753-758 (1973) · Zbl 0272.34056
[14] Pugh, C.: On a theorem of P. Hartman. Amer. J. Math. 91, 363-367 (1969) · Zbl 0197.20701
[15] Sell, G.: Smooth linearization near a fixed point. Amer. J. Math. 107, 1035-1091 (1985) · Zbl 0574.34025
[16] Sternberg, S.: Local contractions and a theorem of Poincaré. Amer. J. Math. 79, 809-824 (1957) · Zbl 0080.29902
[17] Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space. II. Amer. J. Math. 80, 623-631 (1958) · Zbl 0083.31406