= 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 by
$$
b(tf^n)=b_0(t)g^n,\qquad x\le t<xf,\quad n\in\mathbb Z.
$$
The 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.
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