Topics in the theories of means and discrete dynamical systems

Abstract

In Part I, we study means, and prove new results about them. In particular, we define and study the translation means, we prove new results about Gauss composition of means, and we define and study families of means. In Part II, we study discrete dynamical system. We enhance the proof of a theorem about random discrete dynamical system. Further, we study generalisations of dynamical systems related to the Collatz conjecture, in our analysis of which we use the translation mean.