 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.