626 research outputs found
A semi-invertible Oseledets Theorem with applications to transfer operator cocycles
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with
the exponential growth rates of vectors under the action of a linear cocycle on
R^d. When the linear actions are invertible, the MET guarantees an
almost-everywhere pointwise splitting of R^d into subspaces of distinct
exponential growth rates (called Lyapunov exponents). When the linear actions
are non-invertible, Oseledets' MET only yields the existence of a filtration of
subspaces, the elements of which contain all vectors that grow no faster than
exponential rates given by the Lyapunov exponents. The authors recently
demonstrated that a splitting over R^d is guaranteed even without the
invertibility assumption on the linear actions. Motivated by applications of
the MET to cocycles of (non-invertible) transfer operators arising from random
dynamical systems, we demonstrate the existence of an Oseledets splitting for
cocycles of quasi-compact non-invertible linear operators on Banach spaces.Comment: 26 page
A concise proof of the Multiplicative Ergodic Theorem on Banach spaces
We give a streamlined proof of the multiplicative ergodic theorem for
quasi-compact operators on Banach spaces with a separable dual.Comment: 18 page
A spectral approach for quenched limit theorems for random expanding dynamical systems
We prove quenched versions of (i) a large deviations principle (LDP), (ii) a
central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for
non-autonomous dynamical systems. A key advance is the extension of the
spectral method, commonly used in limit laws for deterministic maps, to the
general random setting. We achieve this via multiplicative ergodic theory and
the development of a general framework to control the regularity of Lyapunov
exponents of \emph{twisted transfer operator cocycles} with respect to a twist
parameter. While some versions of the LDP and CLT have previously been proved
with other techniques, the local central limit theorem is, to our knowledge, a
completely new result, and one that demonstrates the strength of our method.
Applications include non-autonomous (piecewise) expanding maps, defined by
random compositions of the form . An important aspect of our results is that we
only assume ergodicity and invertibility of the random driving
; in particular no expansivity or mixing properties areComment: revised version taking into account referee's comments. Accepted for
publication in Communications in Mathematical Physic
- …