Abstract
As already noted in the preface, the main objective of this book is to give a fairly broad overview of the relation between admissibility and hyperbolicity. In this chapter we describe in a pragmatic manner the origins of the theory and we give a brief overview of the contents of the book.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 1–235 (1969)
D. Anosov, Y. Sinai, Certain smooth ergodic systems. Russ. Math. Surv. 22, 103–167 (1967)
L. Barreira, Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and its Applications, vol. 115 (Cambridge University Press, Cambridge, 2007)
C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70 (American Mathematical Society, Providence, 1999)
C. Coffman, J. Schäffer, Dichotomies for linear difference equations. Math. Ann. 172, 139–166 (1967)
W. Coppel, Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629 (Springer, New York, 1981)
J. Dalec’kiı̆, M. Kreı̆n, Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 43 (American Mathematical Society, Providence, RI, 1974)
J. Hale, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25 (American Mathematical Society, Providence, RI, 1988)
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840 (Springer, Berlin, 1981)
N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354 (2006)
B. Levitan, V. Zhikov, Almost Periodic Functions and Differential Equations (Cambridge University Press, Cambridge, 1982)
T. Li, Die Stabilitätsfrage bei Differenzengleichungen. Acta Math. 63, 99–141 (1934)
J. Massera, J. Schäffer, Linear differential equations and functional analysis. I. Ann. of Math. (2) 67, 517–573 (1958)
J. Massera, J. Schäffer, Linear Differential Equations and Function Spaces. Pure and Applied Mathematics, vol. 21 (Academic, New York, 1966)
J. Mather, Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479–483 (1968)
G. Papaschinopoulos, Dichotomies in terms of Lyapunov functions for linear difference equations. J. Math. Anal. Appl. 152, 524–535 (1990)
O. Perron, Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703–728 (1930)
A. Sasu, Exponential dichotomy and dichotomy radius for difference equations. J. Math. Anal. Appl. 344, 906–920 (2008)
G. Sell, Y. You, Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol. 143 (Springer, New York, 2002)
V. Slyusarchuk, Exponential dichotomy for solutions of discrete systems. Ukr. Math. J. 35, 98–103 (1983)
S. Smale, Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Barreira, L., Dragičević, D., Valls, C. (2018). Introduction. In: Admissibility and Hyperbolicity. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-90110-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-90110-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90109-1
Online ISBN: 978-3-319-90110-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)