# 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 and Continuous Dynamical Systems, 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), 791-811. [2] L. Barreira, D. Dragičević and C. Valls, Admissibility in the strong and weak senses,, preprint., (). [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23, Amer. Math. Soc., 2002. [4] L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026. [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations, 228 (2006), 285-310. doi: 10.1016/j.jde.2006.04.001. [6] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241. doi: 10.1007/s10884-006-9026-1. [7] L. Barreira and C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations, 232 (2007), 675-701. doi: 10.1016/j.jde.2006.09.021. [8] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes in Math., 1926, Springer, 2008. doi: 10.1007/978-3-540-74775-8. [9] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies, J. Differential Equations, 246 (2009), 183-215. doi: 10.1016/j.jde.2008.06.009. [10] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999. doi: 10.1090/surv/070. [11] C. Coffman and J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. doi: 10.1007/BF01350095. [12] W. Coppel, On the stability of ordinary differential equations, J. London Math. Soc., 39 (1964), 255-260. [13] W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math., 629, Springer, 1978. [14] Ju. Dalec$'$kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., 840, Springer, 1981. [16] N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002. [17] N. Huy and N. Ha, Exponential dichotomy of difference equations in $l_p$-phase spaces on the half-line, Adv. Difference Equ., 2006, Art. ID 58453, 14 pp. [18] Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414. [19] Yu. Latushkin and Yu. Tomilov, Fredholm differential operators with unbounded coefficients, J. Differential Equations, 208 (2005), 388-429. doi: 10.1016/j.jde.2003.10.018. [20] B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982. [21] T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141. doi: 10.1007/BF02547352. [22] A. Maizel', On stability of solutions of systems of differential equations, Trudi Ural'skogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50. [23] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2), 67 (1958), 517-573. doi: 10.2307/1969871. [24] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Academic Press, 1966. [25] K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [26] K. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [27] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. [28] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261-1305. [29] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. [30] S. Pilyugin, Generalizations of the notion of hyperbolicity, J. Difference Equ. Appl., 12 (2006), 271-282. doi: 10.1080/10236190500489350. [31] V. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations (in Russian), in Problems of Asymptotic Theory of Nonlinear Oscillations, Naukova Dumka, Kiev, 1977, 168-173. [32] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [33] D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187-4205. doi: 10.3934/dcds.2013.33.4187.

show all references

##### References:
 [1] L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility, Adv. Nonlinear Stud., 14 (2014), 791-811. [2] L. Barreira, D. Dragičević and C. Valls, Admissibility in the strong and weak senses,, preprint., (). [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23, Amer. Math. Soc., 2002. [4] L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026. [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations, 228 (2006), 285-310. doi: 10.1016/j.jde.2006.04.001. [6] L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241. doi: 10.1007/s10884-006-9026-1. [7] L. Barreira and C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations, 232 (2007), 675-701. doi: 10.1016/j.jde.2006.09.021. [8] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes in Math., 1926, Springer, 2008. doi: 10.1007/978-3-540-74775-8. [9] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies, J. Differential Equations, 246 (2009), 183-215. doi: 10.1016/j.jde.2008.06.009. [10] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999. doi: 10.1090/surv/070. [11] C. Coffman and J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. doi: 10.1007/BF01350095. [12] W. Coppel, On the stability of ordinary differential equations, J. London Math. Soc., 39 (1964), 255-260. [13] W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math., 629, Springer, 1978. [14] Ju. Dalec$'$kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., 840, Springer, 1981. [16] N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002. [17] N. Huy and N. Ha, Exponential dichotomy of difference equations in $l_p$-phase spaces on the half-line, Adv. Difference Equ., 2006, Art. ID 58453, 14 pp. [18] Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414. [19] Yu. Latushkin and Yu. Tomilov, Fredholm differential operators with unbounded coefficients, J. Differential Equations, 208 (2005), 388-429. doi: 10.1016/j.jde.2003.10.018. [20] B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982. [21] T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141. doi: 10.1007/BF02547352. [22] A. Maizel', On stability of solutions of systems of differential equations, Trudi Ural'skogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50. [23] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2), 67 (1958), 517-573. doi: 10.2307/1969871. [24] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Academic Press, 1966. [25] K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [26] K. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [27] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. [28] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261-1305. [29] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. [30] S. Pilyugin, Generalizations of the notion of hyperbolicity, J. Difference Equ. Appl., 12 (2006), 271-282. doi: 10.1080/10236190500489350. [31] V. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations (in Russian), in Problems of Asymptotic Theory of Nonlinear Oscillations, Naukova Dumka, Kiev, 1977, 168-173. [32] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [33] D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187-4205. doi: 10.3934/dcds.2013.33.4187.
 [1] Luis Barreira, Claudia Valls. Nonuniform exponential dichotomies and admissibility. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 39-53. doi: 10.3934/dcds.2011.30.39 [2] Luis Barreira, Claudia Valls. Noninvertible cocycles: Robustness of exponential dichotomies. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4111-4131. doi: 10.3934/dcds.2012.32.4111 [3] Christian Pötzsche. Smooth roughness of exponential dichotomies, revisited. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 853-859. doi: 10.3934/dcdsb.2015.20.853 [4] Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025 [5] Davor Dragičević. Admissibility and polynomial dichotomies for evolution families. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1321-1336. doi: 10.3934/cpaa.2020064 [6] César M. Silva. Admissibility and generalized nonuniform dichotomies for discrete dynamics. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3419-3443. doi: 10.3934/cpaa.2021112 [7] Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure and Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278 [8] Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246 [9] Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164 [10] Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148 [11] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [12] Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857 [13] Peng Sun. Exponential decay of Lebesgue numbers. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773 [14] José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329 [15] Suzana Antunović, Tonči Kokan, Tanja Vojković, Damir Vukičević. Exponential generalised network descriptors. Advances in Mathematics of Communications, 2019, 13 (3) : 405-420. doi: 10.3934/amc.2019026 [16] Gennady Bachman. Exponential sums with multiplicative coefficients. Electronic Research Announcements, 1999, 5: 128-135. [17] Emmanuel Schenck. Exponential gaps in the length spectrum. Journal of Modern Dynamics, 2020, 16: 207-223. doi: 10.3934/jmd.2020007 [18] Assane Lo, Nasser-eddine Tatar. Exponential stabilization of a structure with interfacial slip. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6285-6306. doi: 10.3934/dcds.2016073 [19] Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219 [20] A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

2020 Impact Factor: 1.392