Conjugating positive bumps while preserving a fundamental interval

Let $f,g$ be positive bumps of an order automorphism whose supports of a permutation have matching endpoint types, as required by conjugacy of positive real-line bumps. Suppose $c<d<\min \{cf,cg\}$, using right actions. Their conjugacy can be chosen to fix $[c,d]$ pointwise.

Start with an increasing bijection $b_0:[c,cf]\to [c,cg]$ that is the identity on $[c,d]$. Such an extension exists because both right endpoints lie strictly beyond $d$. Extend $b_0$ along the bump iterates using $b(t{f}^n)=b_0(t)g^n$, and extend over the complementary rays. Endpoint agreement makes the iterated pieces fit, and gives $fb=bg$ while retaining the prescribed fixed interval.

There is also a useful commuting version. If both bump supports lie strictly inside one fundamental interval of another order automorphism $h$ moving points to the right, choose the conjugator within that cell and fix its endpoints. Repeat it on all $h$-translates of the cell, and use the identity outside the resulting orbital. The pieces define a conjugator commuting with $h$. It also commutes with every automorphism whose support lies in an interval that the conjugator fixes pointwise.