# American Institute of Mathematical Sciences

May  2015, 20(3): 853-859. doi: 10.3934/dcdsb.2015.20.853

## Smooth roughness of exponential dichotomies, revisited

 1 Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  October 2013 Revised  April 2014 Published  January 2015

As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
Citation: Christian Pötzsche. Smooth roughness of exponential dichotomies, revisited. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 853-859. doi: 10.3934/dcdsb.2015.20.853
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##### References:
 [1] Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 [2] Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383 [3] Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457 [4] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 [5] Lei Wang, Jinlong Yuan, Yingfang Li, Enmin Feng, Zhilong Xiu. Parameter identification of nonlinear delayed dynamical system in microbial fermentation based on biological robustness. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 103-113. doi: 10.3934/naco.2014.4.103 [6] Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 [7] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215 [8] Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190 [9] Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283 [10] Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649 [11] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [12] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [13] Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443 [14] Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727 [15] David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96 [16] Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065 [17] Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012 [18] Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855 [19] Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041 [20] John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515

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