Expansions of o-minimal structures on the real field by trajectories of linear vector fields

Abstract:

Let $\mathfrak R$ be an o-minimal expansion of the field of real numbers that defines nontrivial arcs of both the sine and exponential functions. Let $\mathcal G$ be a collection of images of solutions on intervals to differential equations $y’=F(y)$, where $F$ ranges over all $\mathbb R$-linear transformations $\mathbb R^n\to \mathbb R^n$ and $n$ ranges over $\mathbb N$. Then either the expansion of $\mathfrak R$ by the elements of $\mathcal G$ is as well behaved relative to $\mathfrak R$ as one could reasonably hope for or it defines the set of all integers $\mathbb Z$ and thus is as complicated as possible. In particular, if $\mathfrak R$ defines any irrational power functions, then the expansion of $\mathfrak R$ by the elements of $\mathcal G$ either is o-minimal or defines $\mathbb Z$.