In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

Preface. 1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds. 2. Hyperbolic Sets of Diffeomorphisms. 3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets. 4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms. 5. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism. 6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and asymptotic Phase. 7. Hyperbolic Sets of Ordinary Differential Equations. 8. Transversal Homoclinic Points and Hyperbolic Sets in Differential Equations. 9. Shadowing Theorems for Hyperbolic Sets of Differential Equations. 10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations. 11. Numerical Shadowing. References.