Introduction to Dynamical Systems

Abstract

This 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.

Time evolution of Wikipedia network ranking

“…The hyperlinks between Wikipedia articles represent a directed network which reminds a structure of the World Wide Web (WWW). Hence, the mathematical tools developed for WWW search engines, based on the Markov chains [4], Perron-Frobenius operators [5] and the PageRank algorithm of the corresponding Google matrix [6,7], give solid mathematical grounds for analysis of information flow on the Wikipedia network. In this work we perform the Google matrix analysis of Wikipedia network of English articles extending the results presented in [8][9][10][11].…”

An Informational Proof of H-Theorem

“…Topological entropy [20] is defined, following a similar idea, as the logarithm of a suitable set cardinality, but according to a limit process making hard its computational evaluation. New notions of structural entropies over finite structures, with efficient algorithms for computing them, are surely of great interest in all the cases where classical information entropy requires to be modified, for coping with the specific needs of many applicative contexts [21].…”

Characterising porosity of multi-component mixtures in rotary mills

“…This summary is based on Walters (1982) and Brin and Stuck (2002). Consider a set X, and a σ-algebra β of subsets of X. β is a non-empty collection of subsets of set X closed under complements and countable unions (and intersections), where (1) , µ) is called a probability space and µ is called a probability measure.…”