The Multiplicative Ergodic Theorem on Bundles and Manifolds
This chapter deals with extensions of the MET from matrix cocycles in ℝ d to linear cocycles on bundles, in particular to the linearization of a nonlinear RDS on the tangent bundle, by which we prepare the smooth ergodic theory of Part III.
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Summary This chapter deals with extensions of the MET from matrix cocycles in ~d to linear cocycles on bundles, in particular to the linearization of a nonlinear RDS on the tangent bundle, by which we prepare the smooth ergodic theory of Part III. In order to facilitate the transfer of data of the MET from one bundle to another we need a "tempered" version of coordinate change called Lyapunov cohomology (Sect. 4.1). In Sect. 4.2 the MET for the linearization of a nonlinear smooth RDS on a manifold is derived (Theorem 4.2.6). In case the RDS is generated by an RDE or SDE, we also give criteria in terms of the vector fields and the invariant measure ensuring the validity of the integrability conditions of the MET (Theorem 4.2.10 for the RDE case, Theorems 4.2.12, 4.2.13 and 4.2.14 for the SDE case). Sect. 4.3 is devoted to one of the most important techniques of smooth ergodic theory, namely the use of (random) norms which "eat up" the nonuniformity of the MET (Theorem 4.3.6). It is of course crucial that the random norms do not change the Lyapunov exponents (Corollary 4.3.10). We also obtain what is called the "strong version" of the MET (Theorem 4.3.12).
4.1 Tempered Random Variables and Lyapunov Cohomology 4.1.1 Tempered Random Variables We first introduce a concept which is of fundamental importance for most parts of the theory of RDS and can even be considered one of its characteristic features. In many estimates for RDS we will have random "constants" whose values have to be controlled along the orbits of the DS e. For example, the MET for 'll' = Z gives for each s > 0 a finite random variable C10 such that (4.1.1)
L. Arnold, Random Dynamical Systems © Springer-Verlag Berlin Heidelberg 1998
164
Chapter 4. MET on Bundles
There are reasons to consider the norm of if>( -n, w) -l we obtain from (4.1.1) that llif>(n,e-nw)ll :S Cc-(e-nw)e(.A 1 +c-ln,
= if>( n, e-nw) for which n
E z+.
In order not to destroy this estimate we have to exclude exponential growth (at other occasions: exponential decay) of the sequence Cc-(e-nw).
4.1.1 Definition (Tempered Random Variable). (i) A random variable R : n --+ (0, oo) is called tempered with respect to the DS () if for the associated stationary stochastic process t r-+ R( () (t) ·) the invariant set for which . 1 hm -log R(B(t)w)
t-+±oo
t
=0
(4.1.2)
(t--+ -oo applies only to two-sided time) has full JFD-measure. (ii) R : n --+ [0, oo) is called tempered from above if lim -1 log+ R(B(t)w)
t-+±oo
t
1 1
=0
JFD-a.s.,
while R : n --+ (0, oo] is called tempered from below if above, equivalently, if, with log- x := max(O, -logx), 1 lim - I log- R(B(t)w)
t-+±oo
t
1
=0
1i
(4.1.3) is tempered from
JFD-a. s.
(4.1.4)
(iii) A random variable f : n --+ JR.d is called tempered (from above or below) with respect to the DS () if the stationary stochastic process t r-+ llf(B(t)·)ll is tempered (from above or below). • Since
.
1
.
1
t
t-+±oo
t
hm - log+ R(B(t)w) 2: hmsup - logR(B(t)w) 11 11
t-+±oo
and
1 1 lim - log- R(B(t)w) 2: -liminf - logR(B(t)w), 11 11
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