The dynamics of N-multiplications on the circle and Korenblum-Nevanlinna theory

The classification of atomless probability measures on the unit circle which are invariant under at least two pairwise prime integers is a well known unsolved problem. We construct something more general than measures, namely atomless "premesures of bounded $\kappa$-variation" in the sense of Korenblum which are invariant under s given pairwise prime integers. The relevant function $\kappa$ is a generalized entropy function depending on s. This uses Korenblum's generalized Nevanlinna theory. Passing to "$\kappa$-singular measures" and extending these to elements in a Grothendieck group of possibly unbounded measures on the circle, one obtains generalized invariant measures which are carried by "$\kappa$-Carleson" sets. The range of this construction depends on interesting questions about cyclicty in growth algebras of analytic functions on the unit disc and it contains all invariant measures in the usual sense. If there is time, we also discuss some very formal relations with Witt vectors. For example the Artin-Hasse p-exponential from number theory "is" a p-invariant premeasure of bounded $\kappa_1$ variation" (but not a measure).

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