Dynamical Systems : An Introduction Introduction Basic Notions and Examples Topological Dynamics Low Dimensional Dynamics Hyperbolic Dynamics I Hyperbolic Dynamics II Symbolic Dynamics Ergodic Theory. The theory of dynamical systems is a broad and active research subject with connections to most parts of mathematics. Dynamical Systems: An Introduction undertakes the difficult task to provide a self contained and compact introduction. Topics covered include topological low dimensional hyperbolic and symbolic dynamics as well as a brief introduction to ergodic theory. In particular the authors consider topological recurrence topological entropy homeomorphisms and diffeomorphisms of the circle Sharkovskis ordering the Poincaré Bendixson theory and the construction of stable manifolds as well as an introduction to geodesic flows and the study of hyperbolicity (the latter is often absent in a first introduction). Moreover the authors introduce the basics of symbolic dynamics the construction of symbolic codings invariant measures Poincarés recurrence theorem and Birkhoffs ergodic theorem. The exposition is mathematically rigorous concise and direct: all statements (except for some results from other areas) are proven. At the same time the text illustrates the theory with many examples and 140 exercises of variable levels of difficulty. The only prerequisites are a background in linear algebra analysis and elementary topology. This is a textbook primarily designed for a one semester or two semesters course at the advanced undergraduate or beginning graduate levels. It can also be used for self study and as a starting point for more advanced topics.