Results Math 72 (2017), 125–143
2016 Springer International Publishing
published online November 21, 2016
Results in Mathematics
Nonuniform Spectrum on the Half Line
Luis Barreira, Davor Dragiˇcevi´c, and Claudia Valls
Abstract. For a one-sided nonautonomous dynamics deﬁned by a sequence
of invertible matrices, we develop a spectral theory (in the sense of Sacker
and Sell) for the notion of a nonuniform exponential dichotomy with an
arbitrarily small nonuniform part. We emphasize that this notion is ubiq-
uitous in the context of ergodic theory, unlike the notion of a uniform
exponential dichotomy. In particular, we show that each Lyapunov expo-
nent belongs to one interval of the spectrum. We also consider a class of
suﬃciently small nonlinear perturbations of a linear dynamics satisfying
a nonuniform bounded growth condition and we show that each solution
is either eventually zero or the Lyapunov exponents belong to one interval
of the spectrum.
Mathematics Subject Classiﬁcation. Primary 37D99.
Keywords. Nonuniform hyperbolicity, spectrum, perturbations.
For a one-sided nonautonomous dynamics deﬁned by a sequence of invertible
d × d matrices (A
, we consider the notion of a nonuniform exponential
dichotomy with an arbitrarily small nonuniform part. Our main objective is
to develop a spectral theory (in the sense of Sacker and Sell) with respect to
this notion and to study its relation to the theory of Lyapunov exponents.
The original Sacker–Sell spectrum (see ) was introduced for linear cocycles
over a ﬂow (or, equivalently, for linear skew-product ﬂows) with respect to
the notion of a uniform exponential dichotomy. The underlying ideas where
L. B. and C. V. were supported by FCT/Portugal through UID/MAT/04459/2013. D. D. was
supported by an Australian Research Council Discovery Project DP150100017 and Croatian
Science Foundation under the project IP-2014-09-2285.