Conjugacy of positive real-line bumps

Two positive bumps of an order automorphism are conjugate in the full order automorphism group of the real line exactly when their supports of a permutation have the same endpoint type: both endpoints finite, only the left endpoint finite, only the right endpoint finite, or neither endpoint finite. The actual finite endpoint locations impose no further restriction.

To construct the conjugacy on the supports, choose $x$ in the support of $f$ and $y$ in the support of $g$. Choose an increasing bijection $b_0:[x,xf)\to [y,yg)$ matching the endpoint limits, using the right-action convention $xf=f(x)$. Extend it byThe iterated fundamental intervals partition each support, so this defines an increasing bijection satisfying $fb=bg$. If the endpoint types agree, extend $b$ over the corresponding complementary rays, where both bumps are the identity. This gives a global order automorphism with $b^{-1}fb=g$.

Endpoint type is necessary because a global increasing bijection of the real line preserves finite endpoints and the two directions of infinity. For example, the translation $f(x)=x+1$ has no fixed points, while a positive bump supported on $(0,1)$ fixes every point outside that interval. They cannot be conjugate. Within the bounded-support case, the construction shows that interval length and endpoint positions do not distinguish conjugacy classes.

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.