## Introduction to Dynamical SystemsThis book provides a broad introduction to the subject of dynamical systems, suitable for a one- or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to such areas as number theory, data storage, and Internet search engines. This book grew out of lecture notes from the graduate dynamical systems course at the University of Maryland, College Park, and reflects not only the tastes of the authors, but also to some extent the collective opinion of the Dynamics Group at the University of Maryland, which includes experts in virtually every major area of dynamical systems. |

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### Contents

CHAPTER TWO Topological Dynamics | 28 |

CHAPTER THREE Symbolic Dynamics | 54 |

CHAPTER FOUR Ergodic Theory | 69 |

CHAPTER FIVE Hyperbolic Dynamics | 106 |

CHAPTER SIX Ergodicity of Anosov Diffeomorphisms | 141 |

CHAPTER SEVEN LowDimensional Dynamics | 153 |

CHAPTER EIGHT Complex Dynamics | 191 |

CHAPTER NINE MeasureTheoretic Entropy | 208 |

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absolutely continuous Anosov diffeomorphism attracting periodic point attractor called compact metric space conjugate constant contains continuous map converges COROLLARY critical point deﬁne defined denote dense differentiable distal dynamical systems eigenvalue example Exercise ﬁnite ﬁrst fixed point foliation follows forward orbit graph hence homeomorphism hyperbolic set hyperbolic toral automorphism implies integer intersection interval invariant isometry isomorphic Julia set Lebesgue measure Lemma Let f M¨obius transformation map f Markov matrix measure space measure-preserving transformation metric space neighborhood non-empty non-negative open set partition periodic orbit periodic point point of f point of period preimages Proof Proposition Prove rational map Riemannian satisﬁes sequence Show stable and unstable subset subshift subspace Suppose T-invariant tangent Theorem topological entropy topologically mixing topologically transitive transversely uniquely ergodic unstable manifolds vector vertex weak mixing