Source: Notre Dame J. Formal Logic
Volume 47, Number 1
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.
 Dries, L. van den, "The field of reals with a predicate for the powers of two", Manuscripta Mathematica, vol. 54 (1985), pp. 187--95.
Mathematical Reviews (MathSciNet): MR808687
 Dries, L. van den, "o-minimal structures", pp. 137--85 in Logic: From Foundations to Applications (Staffordshire, 1993), edited by W. Hodges, M. Hyland, and C. Steinhorn, Oxford Science Publications, Oxford University Press, New York, 1996.
 Dries, L. van den, "$T$"-convexity and tame extensions. II", The Journal of Symbolic Logic, vol. 62 (1997), pp. 14--34.
 Friedman, H., and C. Miller, "Expansions of o-minimal structures by fast sequences", The Journal of Symbolic Logic, vol. 70 (2005), pp. 410--18.
 Miller, C., "Avoiding the projective hierarchy in expansions of the real field by sequences", forthcoming in Proceedings of the American Mathematical Society.
 Miller, C., "Exponentiation is hard to avoid", Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 257--59.
 Miller, C., "Expansions of dense linear orders with the intermediate value property", The Journal of Symbolic Logic, vol. 66 (2001), pp. 1783--90.
 Miller, C., "Tameness in expansions of the real field", pp. 281--316 in Logic Colloquium '01, vol. 20 of Lecture Notes in Logic, edited by M. Baaz, S. D. Friedman, and J. Krajicek, Association for Symbolic Logic/A. K. Peters, Urbana/Natick, 2005.
 Peterzil, Y., and S. Starchenko, "A trichotomy theorem for o-minimal structures", Proceedings of the London Mathematical Society, Third Series, vol. 77 (1998), pp. 481--523.
 Tyne, J., $T$-levels and $T$-convexity, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2003.