17 Giugno, 2009 17:00

Seminario Matematico e Fisico di Milano

Analysis of the flow associated with the function $n\mapsto {\lambda }^{n^k}$

Isaac Namioka, University of Washington

Dipartimento di Matematica, Politecnico di Milano, Aula Seminari III piano

A flow is a pair $(S,X)$, where $X$ is a compact Hausdorff space and $S$ is a semigroup of continuous
maps of $X$ into itself. In this joint work with Ali Jabbari, we analyze the family of special flows.
Let $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\},k\in \mathbb{N}$ and $\lambda =e^{2\pi it}$ with $t$ a
fixed irrational real number and let $f\in {\mathbb{T}}^{\mathbb{Z}}$ be defined by $f(n)={\lambda }^{n^k}$.
As usual we define the shift operator $U:{\mathbb{T}}^{\mathbb{Z}}\to {\mathbb{T}}^{\mathbb{Z}}$
by $Ug(n)=g(n+1)$ for each $g\in {\mathbb{T}}^{\mathbb{Z}}$.
Let $X_f$ be the closure of the orbit $\{U^{n}f:n\in \mathbb{N}\}$.
Then the flow we are interested is of the form $(\{U^n:n\in \mathbb{N}\},X_f)$.
Previously the case $k=2$ was treated by us in 1984, and the case $k=4$ was considered by P.Milnes.