Transitions and Anti-integrable Limits for Multi-hole Sturmian Systems and Denjoy Counterexamples

For a Denjoy homeomorphism f of the circle S, we call a pair of distinct points of the ω-limit set ω(f) whose forward and backward orbits converge together a gap, and call an orbit of gaps a hole. In this paper, we generalise the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalised Sturmian system. Moreover, for any Denjoy homeomorphism f having a finite number of holes and for any transitive orientation-preserving homeomorphism f1 of the circle with the same rotation number ρ(f1) as ρ(f), we construct a family f of Denjoy homeomorphisms of rotation number ρ(f) containing f such that (ω(f ), f ) is conjugate to (ω(f), f) for 0 < < ̃ < 1 but the number of holes changes at = ̃, that (ω(f ), f ) is conjugate to (ω(f̃), f̃) for ̃ ≤ < 1 but lim ↗1 f (t) = f1(t) for any t ∈ S, and that f has a singular limit when ↘ 0. We show this singular limit is an anti-integrable limit in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The anti-integrable limit can be degenerate or non-degenerate. All transitions can be precisely described in terms of the generalised Sturmian systems.