# American Institute of Mathematical Sciences

July  2015, 35(7): 2817-2844. doi: 10.3934/dcds.2015.35.2817

## From one-sided dichotomies to two-sided dichotomies

 1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa 2 Department of Mathematics, University of Rijeka, 51000 Rijeka, Croatia

Received  March 2014 Revised  November 2014 Published  January 2015

For a general nonautonomous dynamics on a Banach space, we give a necessary and sufficient condition so that the existence of one-sided exponential dichotomies on the past and on the future gives rise to a two-sided exponential dichotomy. The condition is that the stable space of the future at the origin and the unstable space of the past at the origin generate the whole space. We consider the general cases of a noninvertible dynamics as well as of a nonuniform exponential dichotomy and a strong nonuniform exponential dichotomy (for the latter, besides the requirements for a nonuniform exponential dichotomy we need to have a minimal contraction and a maximal expansion). Both notions are ubiquitous in ergodic theory. Our approach consists in reducing the study of the dynamics to one with uniform exponential behavior with respect to a family of norms and then using the characterization of uniform hyperbolicity in terms of an admissibility property in order to show that the dynamics admits a two-sided exponential dichotomy. As an application, we give a complete characterization of the set of Lyapunov exponents of a Lyapunov regular dynamics, in an analogous manner to that in the Sacker--Sell theory.
Citation: Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
##### References:
 [1] L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility,, Adv. Nonlinear Stud., 14 (2014), 791. Google Scholar [2] L. Barreira, D. Dragičević and C. Valls, Admissibility in the strong and weak senses,, preprint., (). Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (2002). Google Scholar [4] L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential Equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. Google Scholar [6] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity,, J. Dynam. Differential Equations, 19 (2007), 215. doi: 10.1007/s10884-006-9026-1. Google Scholar [7] L. Barreira and C. Valls, Stability theory and Lyapunov regularity,, J. Differential Equations, 232 (2007), 675. doi: 10.1016/j.jde.2006.09.021. Google Scholar [8] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations,, Lect. Notes in Math., (1926). doi: 10.1007/978-3-540-74775-8. Google Scholar [9] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies,, J. Differential Equations, 246 (2009), 183. doi: 10.1016/j.jde.2008.06.009. Google Scholar [10] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations,, Mathematical Surveys and Monographs, (1999). doi: 10.1090/surv/070. Google Scholar [11] C. Coffman and J. Schäffer, Dichotomies for linear difference equations,, Math. Ann., 172 (1967), 139. doi: 10.1007/BF01350095. Google Scholar [12] W. Coppel, On the stability of ordinary differential equations,, J. London Math. Soc., 39 (1964), 255. Google Scholar [13] W. Coppel, Dichotomies in Stability Theory,, Lect. Notes. in Math., (1978). Google Scholar [14] Ju. Dalec$'$kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space,, Translations of Mathematical Monographs, (1974). Google Scholar [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes in Math., (1981). Google Scholar [16] N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. Google Scholar [17] N. Huy and N. Ha, Exponential dichotomy of difference equations in $l_p$-phase spaces on the half-line,, Adv. Difference Equ., (2006). Google Scholar [18] Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations,, J. Operator Theory, 58 (2007), 387. Google Scholar [19] Yu. Latushkin and Yu. Tomilov, Fredholm differential operators with unbounded coefficients,, J. Differential Equations, 208 (2005), 388. doi: 10.1016/j.jde.2003.10.018. Google Scholar [20] B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations,, Cambridge University Press, (1982). Google Scholar [21] T. Li, Die Stabilitätsfrage bei Differenzengleichungen,, Acta Math., 63 (1934), 99. doi: 10.1007/BF02547352. Google Scholar [22] A. Maizel', On stability of solutions of systems of differential equations,, Trudi Ural'skogo Politekhnicheskogo Instituta, 51 (1954), 20. Google Scholar [23] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517. doi: 10.2307/1969871. Google Scholar [24] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces,, Pure and Applied Mathematics, (1966). Google Scholar [25] K. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. Google Scholar [26] K. Palmer, Exponential dichotomies and Fredholm operators,, Proc. Amer. Math. Soc., 104 (1988), 149. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar [27] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar [28] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. Google Scholar [29] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. Google Scholar [30] S. Pilyugin, Generalizations of the notion of hyperbolicity,, J. Difference Equ. Appl., 12 (2006), 271. doi: 10.1080/10236190500489350. Google Scholar [31] V. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations (in Russian),, in Problems of Asymptotic Theory of Nonlinear Oscillations, (1977), 168. Google Scholar [32] R. Sacker and G. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. Google Scholar [33] D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory,, Discrete Contin. Dyn. Syst., 33 (2013), 4187. doi: 10.3934/dcds.2013.33.4187. Google Scholar

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##### References:
 [1] L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility,, Adv. Nonlinear Stud., 14 (2014), 791. Google Scholar [2] L. Barreira, D. Dragičević and C. Valls, Admissibility in the strong and weak senses,, preprint., (). Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (2002). Google Scholar [4] L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential Equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. Google Scholar [6] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity,, J. Dynam. Differential Equations, 19 (2007), 215. doi: 10.1007/s10884-006-9026-1. Google Scholar [7] L. Barreira and C. Valls, Stability theory and Lyapunov regularity,, J. Differential Equations, 232 (2007), 675. doi: 10.1016/j.jde.2006.09.021. Google Scholar [8] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations,, Lect. Notes in Math., (1926). doi: 10.1007/978-3-540-74775-8. Google Scholar [9] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies,, J. Differential Equations, 246 (2009), 183. doi: 10.1016/j.jde.2008.06.009. Google Scholar [10] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations,, Mathematical Surveys and Monographs, (1999). doi: 10.1090/surv/070. Google Scholar [11] C. Coffman and J. Schäffer, Dichotomies for linear difference equations,, Math. Ann., 172 (1967), 139. doi: 10.1007/BF01350095. Google Scholar [12] W. Coppel, On the stability of ordinary differential equations,, J. London Math. Soc., 39 (1964), 255. Google Scholar [13] W. Coppel, Dichotomies in Stability Theory,, Lect. Notes. in Math., (1978). Google Scholar [14] Ju. Dalec$'$kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space,, Translations of Mathematical Monographs, (1974). Google Scholar [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes in Math., (1981). Google Scholar [16] N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. Google Scholar [17] N. Huy and N. Ha, Exponential dichotomy of difference equations in $l_p$-phase spaces on the half-line,, Adv. Difference Equ., (2006). Google Scholar [18] Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations,, J. Operator Theory, 58 (2007), 387. Google Scholar [19] Yu. Latushkin and Yu. Tomilov, Fredholm differential operators with unbounded coefficients,, J. Differential Equations, 208 (2005), 388. doi: 10.1016/j.jde.2003.10.018. Google Scholar [20] B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations,, Cambridge University Press, (1982). Google Scholar [21] T. Li, Die Stabilitätsfrage bei Differenzengleichungen,, Acta Math., 63 (1934), 99. doi: 10.1007/BF02547352. Google Scholar [22] A. Maizel', On stability of solutions of systems of differential equations,, Trudi Ural'skogo Politekhnicheskogo Instituta, 51 (1954), 20. Google Scholar [23] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517. doi: 10.2307/1969871. Google Scholar [24] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces,, Pure and Applied Mathematics, (1966). Google Scholar [25] K. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. Google Scholar [26] K. Palmer, Exponential dichotomies and Fredholm operators,, Proc. Amer. Math. Soc., 104 (1988), 149. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar [27] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar [28] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. Google Scholar [29] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. Google Scholar [30] S. Pilyugin, Generalizations of the notion of hyperbolicity,, J. Difference Equ. Appl., 12 (2006), 271. doi: 10.1080/10236190500489350. Google Scholar [31] V. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations (in Russian),, in Problems of Asymptotic Theory of Nonlinear Oscillations, (1977), 168. Google Scholar [32] R. Sacker and G. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. Google Scholar [33] D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory,, Discrete Contin. Dyn. Syst., 33 (2013), 4187. doi: 10.3934/dcds.2013.33.4187. Google Scholar
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