This book presents a comprehensive overview of the theory of iterated maps on the interval as dynamical systems, originally published in 1980. It is a reissue of a classic work by Pierre Collet and Jean-Pierre Eckmann, now reprinted in paperback and as an eBook to ensure continued accessibility. The book is divided into three parts: Motivation and Interpretation, Mathematical Aspects and Proofs, and Properties of One-Parameter Families of Maps. It provides a detailed mathematical analysis of the behavior of continuous maps of an interval into itself, focusing on their dynamics, bifurcations, and scaling properties. The authors emphasize that the study of one-dimensional maps is not merely a mathematical curiosity but a way to understand the behavior of dissipative systems, which are physical systems with some form of "friction." These systems tend toward equilibrium when left alone but can exhibit erratic, aperiodic, or chaotic behavior when driven from the outside. The book explores the mathematical structure of these systems, including their orbits, itineraries, and ergodic properties. It also discusses the universal scaling behavior observed in one-parameter families of maps and its extension to higher-dimensional systems. The authors highlight the importance of understanding the relationship between periodic and aperiodic behavior, which is particularly relevant in the context of chaotic systems. They also discuss the relevance of these findings to various natural sciences, including physics, biology, and economics. The book is written in a way that allows readers to focus on either the mathematical aspects or the physical interpretations, or both. It includes detailed proofs, adaptations of existing results, and a thorough bibliography and index. The work is intended to serve as a resource for mathematicians, physicists, and experimentalists seeking to understand the behavior of complex dynamical systems.This book presents a comprehensive overview of the theory of iterated maps on the interval as dynamical systems, originally published in 1980. It is a reissue of a classic work by Pierre Collet and Jean-Pierre Eckmann, now reprinted in paperback and as an eBook to ensure continued accessibility. The book is divided into three parts: Motivation and Interpretation, Mathematical Aspects and Proofs, and Properties of One-Parameter Families of Maps. It provides a detailed mathematical analysis of the behavior of continuous maps of an interval into itself, focusing on their dynamics, bifurcations, and scaling properties. The authors emphasize that the study of one-dimensional maps is not merely a mathematical curiosity but a way to understand the behavior of dissipative systems, which are physical systems with some form of "friction." These systems tend toward equilibrium when left alone but can exhibit erratic, aperiodic, or chaotic behavior when driven from the outside. The book explores the mathematical structure of these systems, including their orbits, itineraries, and ergodic properties. It also discusses the universal scaling behavior observed in one-parameter families of maps and its extension to higher-dimensional systems. The authors highlight the importance of understanding the relationship between periodic and aperiodic behavior, which is particularly relevant in the context of chaotic systems. They also discuss the relevance of these findings to various natural sciences, including physics, biology, and economics. The book is written in a way that allows readers to focus on either the mathematical aspects or the physical interpretations, or both. It includes detailed proofs, adaptations of existing results, and a thorough bibliography and index. The work is intended to serve as a resource for mathematicians, physicists, and experimentalists seeking to understand the behavior of complex dynamical systems.