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September  2008, 22(3): 509-528. doi: 10.3934/dcds.2008.22.509

Growth rates and nonuniform hyperbolicity

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  June 2007 Revised  May 2008 Published  August 2008

We consider linear equations $v'=A(t)v$ that may exhibit different asymptotic behaviors in different directions. These can be thought of as stable, unstable and central behaviors, although here with respect to arbitrary asymptotic rates $e^{c \rho(t)}$ determined by a function $\rho(t)$, including the usual exponential behavior $\rho(t)=t$ as a very special case. In particular, we consider the notion of $\rho$-nonuniform exponential trichotomy, that combines simultaneously the nonuniformly hyperbolic behavior with arbitrary asymptotic rates. We show that for $\rho$ in a large class of rate functions, any linear equation in block form in a finite-dimensional space, with three blocks having asymptotic rates $e^{c \rho(t)}$ respectively with $c$ negative, zero, and positive, admits a $\rho$-nonuniform exponential trichotomy. We also give explicit examples that cannot be made uniform and for which one cannot take $\rho(t)=t$ without making all Lyapunov exponents infinite. Furthermore, we obtain sharp bounds for the constants that determine the exponential trichotomy. These are expressed in terms of appropriate Lyapunov exponents that measure the growth rate with respect to the function $\rho$.
Citation: Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509
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