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

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 $U:{\mathbb T}^{\mathbb Z}\rightarrow {\mathbb T}^{\mathbb Z}$
by $Ug(n)=g(n+1)$ for each $g\in{\mathbb T}^{\mathbb Z}$.
Let $X_f$ be the closure of the orbit $\{U^nf:n\in\mathbb N\}$.
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.