1. Let be a sequence of compact sets in .

If is compact, does there exist a finite collection of indexes such that ?

Example: Let and for all . Then is compact and any finite union of ‘s is smaller than .

2. Now return to the case that is a compact uniformly hyperbolic invariant subset of for each . If is compact, could it be uniformly hyperbolic for ?

In this case we have a neighborhood containing for each such that admits stable and unstable cones. Now forms an open covering of . So a finitely many of will cover . Then also admits a neighborhood on which we have stable and unstable cones. So is uniformly hyperbolic for .

3. Does the situation become better for hyperbolic case? That is, if is hyperbolic and is compact, is there some finitely many that make the whole ?

**Anosov rigidity**. Let be an invariant set on which a diffeomorphism is nonuniformly hyperbolic. It is compact only if is uniformly hyperbolic on . This phenomenon is called Anosov rigidity.

Theorem. A smooth hyperbolic invariant measure for a diffeomorphism of a compact manifold decomposes into countably many ergodic components of positive measure.

If the unstable foliation extends to a continuous foliation of with smooth leaves and if every local unstable leaf (regardless of its size) eventually expands to a certain uniform size when pushed forward, then each ergodic component can be taken to be an open set. This is referred to as** local ergodicity.**

—————–

Let be a compact metric space and be the collection of nonempty compact subsets of endowed with Hausdorff metric . Then is a compact metric space.

Let and consider its limsup and liminf:

.

.