Let P be the ω-orbit of a point under a unary function definable in an o-minimal expansion ℜ of a densely ordered group. If P is monotonically cofinal in the group, and the compositional iterates of the function are cofinal at +\infty in the unary functions definable in ℜ, then the expansion (ℜ, P) has a number of good properties, in particular, every unary set definable in any elementarily equivalent structure is a disjoint union of open intervals and finitely many discrete sets.
"Expansions of o-Minimal Structures by Iteration Sequences." Notre Dame J. Formal Logic 47 (1) 93 - 99, 2006. https://doi.org/10.1305/ndjfl/1143468314