Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 4 / 4 / i / Solution

Write the two supports as $I_f=(l_f,r_f)$ and $I_g=(l_g,r_g)$. Every point in a support moves strictly right: a positive nonidentity order automorphism has no fixed point inside its one support interval. Choose $u\in I_f$, $v\in I_g$ and an increasing bijection $\theta :[u,uf]\to [v,vg]$ matching endpoints. The half-open intervals $[u{f}^n,u{f}^{n+1})$, for $n\in \mathbb{Z}$, partition $I_f$, because the iterates tend to its endpoints. DefineThe endpoint choices make adjacent pieces fit, giving an increasing bijection $b:I_f\to I_g$ with $fb=bg$. Extend over the two complementary closed rays by increasing maps taking $l_f,r_f$ to $l_g,r_g$. Outside the supports both automorphisms are identity, so $fb=bg$ still holds. Therefore $\boxed{b^{-1}fb=g}$. This gives the conjugacy of positive real-line bumps.