Orbits on a nilmanifold under the action of a polynomial sequence of translations

Abstract

It is known that the closure ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ of the orbit ${\mathop{\hbox{\rm Orb}}\nolimits}_{g}(x)$ of a point $x$ of a compact nilmanifold $X$ under a polynomial sequence $g$ of translations of $X$ is a disjoint finite union of sub-nilmanifolds of $X$. Assume that $g(0)=1_{G}$ and let $A$ be the group generated by the elements of $g$; we show in this paper that for almost all points $x\in X$, the closures ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ are congruent (that is, are translates of each other), with connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ equal to (some of) the connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{A}(x)$.