Hyperbolic Dynamics of Endomorphisms

We present the theory of hyperbolic dynamics of endomorphisms in. Topics covered are hyperbolic sets, stable manifolds, local product structure , shadowing, spectral decomposition and ^-stability. 0. Introduction In this paper we study a smooth mapping f of a manifold M as a dynamical system. We will discuss both semilocal and global dynamical properties of f, but always under some hyperbolicity assumption. The main examples we have in mind are holomorphic endomorphisms of complex projective space P k , k 1 but we will state the results in greater generality. There are many excellent and detailed expositions on diierentiable dynamics, e.g. S], but they usually consider only invertible systems, such as diieomorphisms of a compact manifold. As for noninvertible maps, the attitude seems to be that most results for diieomorphisms continue to hold when interpreted correctly, but it is diicult to nd a detailed written account; the purpose of this paper is to improve upon that. We do not claim that our results are new. Our main references are R] and PS]. The building blocks in hyperbolic dynamics are hyperbolic sets. These are generalizations of hyperbolic xed points, i.e. xed points where the derivative has no eigenvalue of modulus one. For the precise deenition of what it means for a compact, invariant set to be hyperbolic, we refer to section 1, but the deenition involves the set ^ = f(x i) i0 ; x i 2 ; f(x i) = x i+1 g: of histories in. A hyperbolic set has local stable and unstable manifolds at each point; see Theorem 1.2 for details. Another basic feature of hyperbolic sets is persistence under perturbations. This means that if f is hyperbolic on = f and g is close to f, then g has a hyperbolic set g close to f such that ^ fj ^ f and ^ gj ^ g are conjugate. Here ^ f is the shift f((x i)) = (f(x i)). For more details see Proposition 1.4. Note that the sets f and g themselves need not be homeomorphic. Many results on the dynamics near a hyperbolic set are best formulated in terms of ^. With this in mind we introduce the concept of local product structure for ^. The deenition says that if (^ p (i)) i2Z and (^ q (i)) i2Z are orbits in ^ and (^ x (i)) i2Z