Preface 1 Simple Interval Maps and Their Iterations 1.1 Introduction 1.2 The Inverse and hnplicit Function Theorems 1.3 Visualizing from the Graphics oflterations of the Quadratic Map Notes for Chapter 1 2 Total Variations of Iterates of Maps 2.1 The Use of Total Variations as a Measure of Chaos Notes for Chapter 2 3 Ordering among Periods: The Sharkovski Theorem Notes for Chapter 3 4 Bifurcation Theorems for Maps 4.1 The Period-Doubling Bifurcation Theorem 4.2 Saddle-Node Bifurcations 4.3 The Pitchfork Bifurcation 4.4 Hopf Bifurcation Notes for Chapter 4 5 Homoclinicity. LyapunoffExponents 5.1 Homoclinic Orbits 5.2 Lyapunoff Exponents Notes for Chapter 5 6 Symbolic Dynamics, Conjugacy and Shift Invariant Sets 6.1 The Itinerary of an Orbit 6.2 Properties of the shift map �� 6.3 Symbolic Dynamical Systems ��k and ��k+ 6.4 The Dynamics of (��l+����+) and Chaos 6.5 Topological Conjugacy and Semiconjugacy 6.6 Shift Im, ariant Sets 6.7 Construction of Shift Invariant Sets 6.8 Snap-back Repeller as a Shift Invariant Set Notes for Chapter 6 7 The Smale Horseshoe 7.1 The Standard Smale Horseshoe 7.2 The General Horseshoe Notes for Chapter 7 8 Fractals 8.1 Examples of Fractals 8.2 HausdorffDimension and the HausdorffMeasure 8.3 Iterated Function Systems (IFS) Notes fbr Chapter 8 9 Rapid Fluctuations of Chaotic Maps on RN 9.1 Total Variation for Vector-Value Maps 9.2 Rapid Fluctuations of Maps on ]RN 9.3 Rapid Fluctuations of Systems with Quasi-shift Invariant Sets 9.4 Rapid Fluctuations of Systems Containing Topological Horseshoes 9.5 Examples of Applications of Rapid Fluctuatkms Notes for Chapter 9 10 Infinite-dimensional Systems Induced by Continuous-Time Difference Equations 10.1 I3DS 10.2 Rates of Growth of Total Varkations of Iterates