Sharp regularity of linearization for $$C^{1,1}$$ C 1 , 1 hyperbolic diffeomorphisms

Zhang, Wenmeng; Jarczyk, Witold

doi:10.1007/s00208-013-0954-x

Cited by 26 publications

(27 citation statements)

References 30 publications

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“…Notice that in [36] the same claims were proved in neighborhoods having sufficiently small diameters d > 0. The only difference in the present version is that we allow the diameter to be arbitrarily large.…”

Section: Appendix: Global Smooth Linearizationsupporting

confidence: 54%

“…Remark that our result of C 1 linearization, Theorem 2, is obtained in the sense of nonuniform dichotomies. The nonuniformity, depending on the initial time in the nonautonomous system, was not considered in [24,36]. A known result ([5, Section 3]) on linearization with such a nonuniformity is concerning a conjugation with a Hölder continuity.…”

Section: Nonautonomous Smooth Linearizationmentioning

confidence: 99%

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Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy

Dragičević

Zhang

2018

Math. Z.

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In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on R d . Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for C 1 linearization of such a nonautonomous difference equation. .Our theorems improve known results even in the case of uniform strong exponential dichotomies.

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Section: Appendix: Global Smooth Linearizationsupporting

confidence: 54%

Section: Nonautonomous Smooth Linearizationmentioning

confidence: 99%

Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy

Dragičević

Zhang

2018

Math. Z.

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“…We mention the recent important results by M.S. ElBialy [2] and by W. Zhang, W. Zhang and W. Jarczyk [10] on resonant fixed points and on sharp regularity estimates, respectively. We also address to our previous paper [9] for more references and some open problems.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Differentiability with respect to parameters in global smooth linearization

Rodrigues

Solà-Morales

2017

Journal of Differential Equations

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Let XX be a Banach space and T¿:X¿XT¿:X¿X a family of invertible contractions, T¿=L¿+f¿T¿=L¿+f¿, where L¿L¿ is linear and f¿f¿ is nonlinear with f¿(0)=0f¿(0)=0. We give conditions for the existence of a family of global linearization maps H¿H¿, such that View the MathML sourceH¿°T¿°H¿-1=L¿, with a smooth dependence on ¿. The results depend strongly on the choice of some appropriate spaces of maps, adapted norms and the use of a specific fixed point theorem with smooth dependence on parametersPeer ReviewedPostprint (author's final draft

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“…Proof of Theorem For any given small constant

η > 0

, there is a small neighborhood

V_{η} \subset {double-struckR}^{d}

of the origin such that one can use a smooth cut‐off function defined in

R^{d}

(that is, a smooth function which is equal to 1 in

V_{η}

and is equal to 0 outside a neighborhood of

V_{η}

) to extend the locally defined

C^{1, 1}

map

f

to a global one satisfying

\begin{matrix} true \\ right ∥ D f false(x false) ∥ ⩽ η and ∥ D f false(x false) - D f false(y false) ∥ ⩽ B ∥ x - y ∥, \forall x, y \in R^{n}, \end{matrix}

where

B > 0

is a constant (see, for example, ). Note that the diameter of

V_{η}

tends to 0 when

η

tends to 0.…”

Section: Simultaneously Differentiable and Hölder Linearizationmentioning

confidence: 99%

“…In 1970s Belitskii gave conditions on

C^{k}

linearization for

C^{k, 1}

(

k ⩾ 1

) diffeomorphisms, which implies that

C^{1, 1}

diffeomorphisms can be

C^{1}

linearized locally if the eigenvalues

λ_{1}, \dots, λ_{n}

satisfy a nonresonant condition that

\begin{matrix} true \\ right | λ_{i} | \cdot | λ_{j} | \neq | λ_{ι} | \end{matrix}

for all

ι = 1, \dots, n

false| λ_{i} false| < 1 < false| λ_{j} false|

. This result was partially generalized to infinite‐dimensional spaces in . Note that in the contractive (or expansive) case holds automatically and therefore

C^{1}

linearization can always be realized in

R^{n}

.…”

Section: Introductionmentioning

confidence: 99%

Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy

Dragičević

Zhang

2019

Proc. London Math. Soc.

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In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of a discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted to the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any nonresonant conditions.

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Sharp regularity of linearization for $$C^{1,1}$$ C 1 , 1 hyperbolic diffeomorphisms

Cited by 26 publications

References 30 publications

Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy

Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy

Differentiability with respect to parameters in global smooth linearization

Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy

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