Nonuniformly hyperbolic cocycles: admissibility and robustness
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 545-564.

We give a relatively short proof of the robustness of nonuniformly hyperbolic cocycles in a Banach space under sufficiently small perturbations. In strong contrast to former proofs, we do not need to construct projections leading to the stable and unstable subspaces. Instead, these are obtained fairly explicitly depending only on the boundedness respectively of forward and backward orbits. A difficulty is that we need to construct from the beginning appropriate sequences of Lyapunov norms, with respect to which one can measure the boundedness of the orbits. These norms need not only to be guessed a priori but also all the computations would change if these were not appropriate, both for the original and for the perturbed cocycles. The proof of the robustness is based on the relation between the notions of nonuniform exponential dichotomy and of admissibility, together with nontrivial norm bounds for the expansion and contraction and for the norms of the projections. This relation allows us to construct an invertible operator from the set of bounded perturbations to the set of bounded solutions, and thus to conclude that under sufficiently small perturbations a similar operator exists for the perturbed cocycle.

Published online:
Classification: 34D09, 37D25
Barreira, Luis 1; Valls, Claudia 1

1 Dep. Matemática Instituto Superior Técnico 1049-001 Lisboa, Portugal
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Barreira, Luis; Valls, Claudia. Nonuniformly hyperbolic cocycles: admissibility and robustness. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 545-564. http://archive.numdam.org/item/ASNSP_2012_5_11_3_545_0/

[1] L. Barreira and Ya. Pesin, “Nonuniform Hyperbolicity”, Encyclopedia of Mathematics and Its Application 115, Cambridge University Press, 2007. | MR | Zbl

[2] L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations 244 (2008), 2407–2447. | MR | Zbl

[3] L. Barreira and C. Valls, Robustness of general dichotomies, J. Funct. Anal. 257 (2009), 464–484. | MR | Zbl

[4] C. Chicone and Yu. Latushkin, “Evolution Semigroups in Dynamical Systems and Differential Equations”, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999. | MR | Zbl

[5] S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), 429–477. | MR | Zbl

[6] W. Coppel, Dichotomies and reducibility, J. Differential Equations 3 (1967), 500–521. | MR | Zbl

[7] Ju. Dalecʼkiĭ and M. Kreĭn, “Stability of Solutions of Differential Equations in Banach Space”, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974. | MR | Zbl

[8] N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), 330–354. | MR | Zbl

[9] B. Levitan and V. Zhikov, “Almost Periodic Functions and Differential Equations”, Cambridge University Press, 1982. | MR | Zbl

[10] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573. | EuDML | MR | Zbl

[11] J. Massera and J. Schäffer, “Linear Differential Equations and Function Spaces”, Pure and Applied Mathematics 21, Academic Press, 1966. | MR | Zbl

[12] M. Megan, B. Sasu and A. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 44 (2002), 71–78. | MR | Zbl

[13] N. V. Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl. 261 (2001), 28–44. | MR | Zbl

[14] N. V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory 32 (1998), 332–353. | MR | Zbl

[15] R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal. 31 (1998), 559–571. | MR | Zbl

[16] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728. | EuDML | JFM | MR

[17] V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (1999), 471–513. | MR | Zbl

[18] L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl. 314 (2006), 436–454. | MR | Zbl

[19] P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc. 27 (1983), 31–52. | MR | Zbl

[20] P. Preda, A. Pogan and C. Preda, (L p ,L q )-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory 49 (2004), 405–418. | MR | Zbl

[21] P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Differential Equations 212 (2005), 191–207. | MR | Zbl

[22] P. Preda, A. Pogan and C. Preda, Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differential Equations 230 (2006), 378–391. | MR | Zbl

[23] A. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl. 344 (2008), 906–920. | MR | Zbl