Chapter 1 Overture

Sect. 1.1 together with the appendix A o ers a historical overview of the development of the theory of chaotic dynamics, mostly of scholarly and human interest only. The book proper starts in sect. 1.3. A pinball game is used to motivate and illustrate most of the concepts to be developed in this book: unstable dynamical ows, Poincar e sections, Smale horseshoes, symbolic dynamics, pruning, discrete symmetries, periodic orbits, averaging over chaotic sets, evolution operators, dynamical zeta functions, spectral determinants, cycle expansions, quantum trace formulas and zeta functions, and so on to the semiclassical quantization of helium. This chapter is a quick parcourse of the main topics covered in the book.