The adic realization of the Morse transformation and the extension of its action to the solenoid

Abstract

We consider the adic realization of the Morse transformation on the additive group of integer dyadic numbers. We discuss the arithmetic properties of this action. Then we extend this action to an action of the group of rational dyadic numbers on the solenoid. Bibliography: 14 titles. To the memory of Alexander Livshits Sasha Livshits (1950{2008) was the author of one of the most important theorems of modern dynami s, whi h is well-known now, | the theorem about the ohomology of hyperboli systems. He proved this theorem when he was a student. Later he worked on many other problems of symboli dynami s, ergodi theory, and ombinatori s. His deep and important ideas made a great impression on those who intera ted with him (this in ludes the se ond author). The rst author onsiders him the best of his students. 1