Infinite ergodic theory of numbers (Q2797267)

Classical ergodic theory is concerned with the measure-theoretic behavior of dynamical systems which preserve a probability measure, that is, a normalized \textit{finite} measure. In contrast, \textit{infinite} ergodic theory is concerned with dynamical systems which preserve an \textit{infinite} measure, such as the Gauss measure in the context of the continued fraction expansion. The present book is intended as an introduction to the field of infinite ergodic theory from a number-theoretic perspective, suitable for advanced undergraduate students or PhD students, and could for example be used for a reading seminar. Some knowledge from measure theory and a bit of functional analysis is required, but apart from that the book is largely self-contained. Among the topics contained in the book are the Gauss map and continued fractions, metric Diophantine approximation, the Farey map, the Stern-Brocot sequence and Minkowski's question mark function, Lüroth expansions, the connection between infinite ergodic theory and renewal theory, notions of mixing, the connection with functional analysis and the Chacon-Ornstein ergodic theorem, and asymptotic results for sum-level sets. The book is written in an accessible way, and there are exercises at the end of each section.