EXPANSIONS OF O-MINIMAL STRUCTURES ON THE REAL FIELD BY TRAJECTORIES OF LINEAR VECTOR FIELDS

摘要

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