Conjugacy of positive real-line bumps

Choose fundamental intervals $[x,xf)$ and $[y,yg)$ for two positive bumps of an order automorphism. Any increasing bijection matching their endpoints extends by $b(x{f}^n)=(xb)g^n$ to their support intervals. Extend it monotonically over the two complementary rays. Then $fb=bg$, so $b^{-1}fb=g$. Bounded supports impose no further invariant for positive bumps in the full real-line order-automorphism group.

If two positive bumps of an order automorphism move a point $c$ beyond an interval $[c,d]$, their conjugacy can be chosen to fix that interval: begin with the identity there, complete the increasing map between fundamental intervals, and extend equivariantly along bump iterates. If both bump supports lie in one fundamental interval of another positive motion $h$, fix that cell's endpoints and repeat the conjugator on all its $h$-translates. The result commutes with $h$, and with any automorphism supported in the interval fixed pointwise.