Let be a compact Hausdorff space, be Borel -algebra, and be the collection of Borel Probability measures on . This again is a compact Hausdorff space.

Fo r each there exists a set such that for every .

If , then in the weak-topolgy is equivalent to either one of the following:

For each closed subset , .

For each open subset , .

For every with , .

If is an open set in for all , then is said to be lower semicontinuous. If is an open set in for all , then is said to be upper semicontinuous. If are a family of l.s.c. (or u.s.c.), then so is (or ) since

The Lyapunov exponent of a smooth diffeo is defined as

for each and . It behaves quite good in the measure sense. But in general there is no more information about its topological properties. For examle it may not be semicontinuous.

Let be a diffeo and be a compact -invariant set. Denote

These are -invariant linear subspaces of , and vectors of and are asymptotic to zero, but not necessarily exponentially fast. However, if the two subspaces form a direct sum at every point of , exponential

convergence will follow:

Proposition (Mane, Liao). is hyperbolic if and only if for all .