Preliminaries

Abstract

In this part we review some necessary concepts and results from ergodic theory, which will be frequently used in this monograph.

Throughout this book, M is an m ₀-dimensional, smooth, compact and connected Riemannian manifold without boundary.We use f ∈ C ^r(O,M) to denote a C ^r map from O to M, where O is an open subset of M, and we call f a C ^r endomorphism on M if f ∈ C ^r (M, M). We use T f to denote the tangent map induced by f when r ≥ 1.

For any compact metrizable space X and continuous map T : X → X, We use M _T (X) to denote the set of T-invariant Borel probability measures on X.