Exercise 21 solution: Suppose o(θ, f) is dense in T. Let x_{0} ∈ π^{−1}(θ). As in the solution of Exercise 8(b), eo(x_{0}) is dense in R. Define \overline{H} : eo(x_{0}) → R as in Proposition 2, and H : R → R as in the proof of Proposition 3. If H is not a homeomorphism, pick an interval J on which H is constant. Since eo(x_{0}) is dense in R, pick x_{1}, x_{2} ∈ eo(x_{0}) ∩ J, x_{1} < x_{2}. Then H(x_{1}) < H(x_{2}) by Proposition 2, contradicting the fact H is constant on J. Hence H is a homeomorphism, f is conjugate to r_{α}, and all orbits of f are dense in T.