Noncompact Manifolds

Up to now, we have considered dynamics on compact manifolds with finite measures. In this last part of the book we widen our analysis to include noncompact manifolds and consequently infinite measures.

• Topologically, the analysis extends to cover sigma compact manifolds X – manifolds which can be represented as a countable union of compact sets. In fact (see Section 14.6), they can be represented as a countable union of compact manifolds. As in the compact case, we allow a manifold boundary, which we denote by ∂X. For noncompact manifolds, the notion of an end (roughly, a way of going to infinity) will turn out to be of great importance. This notion will be introduced informally in Chapter 13, and then more formally in Chapter 14.

• Measure theoretically, the manifold X will be endowed with a fixed OU measure μ which can be finite or infinite, but in any case the definition of an OU measure ensures it is sigma finite. This means the space X can be written as a countable union of sets of finite μ-measure. Mainly we will be interested in the case where the OU measure μ is infinite, as the finite measure case resembles the theory developed earlier for compact manifolds. The relation between the ends of the manifold X and the measure μ will be important for the theory we will develop. Some ends will have infinite measure, and those ends of infinite measure will be significant in the theory.