Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 231
Consider a more abstract setting. Let g : X → X be a measurable map and ν
an invariant probability measure. Let A : X → GL(R,k) be a measurable function.
Define for n ≥ 0
An(x) := A(g n−1 (x))...A(x).
These functions satisfy the identity
An+m(x)=An(g m (x))Am(x)
for n,m ≥ 0. We say that the sequence (An) is the multiplicative cocycle over X
generated by A.
The following Oseledec’s multiplicative ergodic theorem is related to the
Kingman’s sub-multiplicative ergodic theorem [KH, W]. It can be seen as a generalization
of the above property of a single square matrix A.
Theorem 1.119 (Oseledec). Let g : X → X,ν and the cocycle (An) be as above. Assume
that ν is ergodic and that log + �A ±1 (x)� are in L 1 (ν). Then there is an integer
m, real numbers χ1 < ··· < χm, and for ν-almost every x, a unique decomposition
of R k into a direct sum of linear subspaces
such that
R k m�
= Ei(x)
i=1
1. The dimension of Ei(x) does not depend on x.
2. The decomposition is invariant, that is, A(x) sends Ei(x) to Ei(g(x)).
3. We have locally uniformly on vectors v in Ei(x) \{0}
1
lim
n→∞ n log�An(x) · v� = χi.
4. For S ⊂{1,...,m}, define ES(x) := ⊕i∈SEi(x).IfS,S ′ are disjoint, then the angle
between ES(x) and E S ′(x) is a tempered function, that is,
1
lim
n→∞ n logsin� �
�∠ ES(g n (x)),ES ′(g n (x)) �� � = 0.
The result is still valid for non-ergodic systems but the constants m and χi should
be replaced with invariant functions. If g is invertible, the previous decomposition
is the same for g −1 where the exponents χi are replaced with −χi. The result is also
valid in the complex setting where we replace R with C and GL(R,k) by GL(C,k).
In this case, the subspaces Ei(x) are complex.
We now come back to a smooth dynamical system g : X → X on a compact
manifold. We assume that the Jacobian J(g) of g associated to a smooth volume
form satisfies 〈ν,logJ(g)〉 > −∞. Under this hypothesis, we can apply Oseledec’s
theorem to the cocycle induced by g on the tangent bundle of X; this allows to
decompose, ν-almost everywhere, the tangent bundle into invariant sub-bundles.