Admissibility, a general type of Lipschitz shadowing and structural stability
Davor Dragičević
Communications on Pure & Applied Analysis 2015, 14(3): 861-880 doi: 10.3934/cpaa.2015.14.861
For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a uniform exponential dichotomy and we characterize it completely in terms of the admissibility of a large class of function spaces. We apply those results to show that structural stability of a diffeomorphism is equivalent to a very general type of Lipschitz shadowing property. Our results extend those in [37] in various directions.
keywords: Admissibility exponential dichotomies shadowing structural stability.
From one-sided dichotomies to two-sided dichotomies
Luis Barreira Davor Dragičević Claudia Valls
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 2817-2844 doi: 10.3934/dcds.2015.35.2817
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.
keywords: strong exponential dichotomies. Exponential dichotomies

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