An update on the topological properties of asynchronous iterations

International audienceAsynchronous iterations have long been studied from a convergence perspective, and significant results have been obtained over the past fifty years, leading to the successful exploitation of these iterations in the context of asynchronous distributed computing. More recently, many advances in the theoretical study of the randomness of such asynchronous iterations have been achieved, and these results have been successfully applied in various areas of IT security in the past decade. The objective of this article is to review these various advances in the study of the disorder of asynchronous iterations, both theoretically and practically, and to present new avenues of research and the latest results. In detail, we will present the link between asynchronous iterations and a certain category of related graphs, and we will deduce a characterization of the chaos of such iterations, as mathematically defined by Devaney, for Lyapounov’s exponent, etc. These results will be put in perspective with those established long ago at the level of the convergence of these asynchronous iterations. New results from measurement theory will then be discussed, and we will then provide an overview of the 1 applications of these results in computer science, focusing in particular on information security and bioinformatics