Holmgren, R. A. A first course in discrete dynamical systems. (English) Zbl 0797.58001 Universitext. New York, NY: Springer-Verlag. 225 p. (1994). Discrete dynamical systems are essentially iterated functions. The book under review is suitable for a one-semester course on discrete dynamical systems. It contains an extensive quantity of interesting exercises. Let us specify its substance by chapters: the concept of the function; topology of the real numbers; periodical points and stable sets at the iteration of real functions, graphical analysis; Sharkovskij’s theorem; dynamical information contained in the derivative of the function, attracting and repelling points; parametrized families of functions and bifurcations (the definition of the bifurcations of parametrized families of functions, bifurcation diagrams, saddle-node, pitchfork, transcritical and period doubling bifurcations); symbolic dynamics and chaos; the logistic function \(h_ r(x) = rx(1 - x)\), \(r > 0\); Newton’s method from the point of view of discrete dynamics, the iterations for quadratic and cubic functions; numerical solution of differential equations, Euler’s method; the dynamics of complex functions, dynamics of the quadratic maps, Newton’s method in the complex plane; Mandelbrot and Julia sets. The Appendix is devoted to computer algorithms (iterating functions, graphical analysis, bifurcation diagrams, Julia and Mandelbrot sets, stable sets of Newton’s method). Reviewer: B.V.Loginov (Ulyanovsk) Cited in 2 ReviewsCited in 30 Documents MSC: 58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 39A12 Discrete version of topics in analysis 37G99 Local and nonlocal bifurcation theory for dynamical systems 37B99 Topological dynamics Keywords:logistic function; iterated functions; discrete dynamical systems; periodical points; stable sets; Sharkovskij’s theorem; parametrized families of functions; bifurcations; symbolic dynamics; chaos; Newton’s method; Julia sets; Mandelbrot sets PDFBibTeX XMLCite \textit{R. A. Holmgren}, A first course in discrete dynamical systems. New York, NY: Springer (1994; Zbl 0797.58001)