Abstract
We introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. The latter refers to the existence of (unique) bounded solutions for any bounded perturbation of the original dynamics. We consider the general case of a nonautonomous dynamics defined by a sequence of linear operators. As a nontrivial application, we establish the robustness of nonuniform exponential dichotomies as well as of strong nonuniform exponential dichotomies, which corresponds to the persistence of these notions under sufficiently small linear perturbations. The relevance of the results stems from the ubiquity of this type of exponential behavior in the context of ergodic theory: for almost all trajectories with nonzero Lyapunov exponents of a measure-preserving diffeomorphism, the derivative cocycle admits a nonuniform exponential dichotomy and in fact a strong nonuniform exponential dichotomy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Barreira and Ya. Pesin. Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series 23, Amer. Math. Soc., (2002).
L. Barreira and Ya. Pesin. NonuniformHyperbolicity. Encyclopedia ofMathematics and Its Application 115, Cambridge University Press (2007).
L. Barreira and C. Valls. Robustness of nonuniform exponential dichotomies in Banach spaces. J. Differential Equation., 244 (2008), 2407–2447.
L. Barreira and C. Valls. Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics 1926, Springer (2008).
A. Ben-Artzi and I. Gohberg. Dichotomy of systems and invertibility of linear ordinary differential operators, in Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl. 56, Birkhäuser, (1992), 90–119.
A. Ben-Artzi, I. Gohberg and M. Kaashoek. Invertibility and dichotomy of differential operators on a half-line. J. Dynam. Differential Equation., 5 (1993), 1–36.
C. Chicone and Yu. Latushkin. Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs 70, Amer. Math. Soc., (1999).
S.-N. Chow and H. Leiva. Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. J. Differential Equation., 120 (1995), 429–477.
W. Coppel. Dichotomies and reducibility. J. Differential Equations, 3 (1967), 500–521.
W. Coppel. Dichotomies in Stability Theory. Lect. Notes in Math. 629, Springer (1978).
Ju. Daleckiĭ and M. Kreĭn. Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs 43, Amer. Math. Soc., (1974).
D. Henry. Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math. 840, Springer (1981).
N. Huy. Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal., 235 (2006), 330–354.
Yu. Latushkin, A. Pogan and R. Schnaubelt. Dichotomy and Fredholm properties of evolution equations. J. Operator Theor., 58 (2007), 387–414.
B. Levitan and V. Zhikov. Almost Periodic Functions and Differential Equations. Cambridge University Press, (1982).
J. Massera and J. Schäffer. Linear differential equations and functional analysis. I. Ann. of Math. (2), 67 (1958), 517–573.
J. Massera and J. Schäffer. Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21, Academic Press, (1966).
N. Minh and N. Huy. Characterizations of dichotomies of evolution equations on the half-line. J. Math. Anal. Appl., 261 (2001), 28–44.
R. Naulin and M. Pinto. Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal., 31 (1998), 559–571.
V. Oseledets. A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19 (1968), 197–221.
K. Palmer. Exponential dichotomies and Fredholm operators. Proc. Amer. Math. Soc., 104 (1988), 149–156.
O. Perron. Die Stabilitätsfrage bei Differentialgleichungen. Math. Z., 32 (1930), 703–728.
Ya. Pesin. Families of invariantmanifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv., 10 (1976), 1261–1305.
V. Pliss and G. Sell. Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dynam. Differential Equation., 11 (1999), 471–513.
L. Popescu. Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl., 314 (2006), 436–454.
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 Theor., 49 (2004), 405–418.
N. Van Minh, F. Räbiger and R. Schnaubelt. Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral Equations Operator Theor., 32 (1998), 332–353.
W. Zhang. The Fredholm alternative and exponential dichotomies for parabolic equations. J. Math. Anal. Appl., 191 (1985), 180–201.
Author information
Authors and Affiliations
Corresponding author
Additional information
L.B. and C.V. were supported by Portuguese funds through FCT: project PEst-OE/EEI/LA0009/2013 (CAMGSD).
D.D. was partly supported by University of Rijeka research grant 13.14.1.2.02.
About this article
Cite this article
Barreira, L., Dragičević, D. & Valls, C. Characterization of strong exponential dichotomies. Bull Braz Math Soc, New Series 46, 81–103 (2015). https://doi.org/10.1007/s00574-015-0085-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-015-0085-y