In Chapter 1, we saw several examples of dynamical systems that were locally linear and had complementary expanding and/or contracting directions: expanding endomorphisms of S1, hyperbolic toral automorphisms, the horseshoe, and the solenoid. In this chapter, we develop the theory of hyperbolic differentiable dynamical systems, which include these examples. Locally, a differentiable dynamical system is well approximated by a linear map – its derivative. Hyperbolicity means that the derivative has complementary expanding and contracting directions.

The proper setting for a differentiable dynamical system is a differentiable manifold with a differentiable map, or flow. A detailed introduction to the theory of differentiable manifolds is beyond the scope of this book. For the convenience of the reader, we give a brief formal introduction to differentiable manifolds in §5.13, and an even briefer informal introduction here.

For the purposes of this book, and without loss of generality (see the embedding theorems in), it suffices to think of a differentiable manifold Mn as an n-dimensional differentiable surface, or submanifold, in ℝN, N > n. The implicit function theorem implies that each point in M has a local coordinate system that identifies a neighborhood of the point with a neighborhood of 0 in ℝn. For each point x on such a surface M ⊂ ℝN, the tangent space TxM ⊂ ℝN is the space of all vectors tangent to M at x.