The orbits of one-dimensional maps -- Bifurcations and the logistic family -- Sharkovsky's theorem -- Dynamics of metric spaces -- Countability, sets of measure zero and the cantor set -- Devaney's definition of chaos -- Conjugacy of dynamical systems -- Singer's theorem -- Conjugacy, fundamental domains and the tent family -- Fractals -- Newton's method for real quadratics and cubics -- Coppel's theorem and a proof of Sharkovsky's theorem -- Real linear transformations, the HeĢnon Map and hyperbolic toral automorphisms -- Elementary complex dynamics -- Examples of substitutions -- Fractals arising from substitutions -- Compactness in metric spaces and an introduction to topological dynamics -- Substitution dynamical systems -- Sturmian sequences and irrational rotations.
Summary:
Publisher Description (unedited publisher data) This undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky's theorem, and the theory of substitutions, and takes special care in covering Newton's method. Mathematica code is available online, so that students can see implementation of many of the dynamical aspects of the text.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.