Ok, I start to understand. For the common reader on this site, it might help specifying that “x5+1/2” refers to the map n \longmapsto \cfrac{5n+1}{2}.
What you describe are conjugacies between functions of the form
T(n) = \left\{\begin{array}{ll} \frac{an+b}{2} &( n \text{ odd})\\ \frac{n}{2} &(n\text{ even}) \end{array}\right.
and
U(n) = \left\{\begin{array}{ll} \frac{n+b}{2} &( n \text{ odd})\\ \frac{an}{2} &(n\text{ even}) \end{array}\right.
where a,b are odd integers (except that, maybe for simplicity, you omitted the “short-cut” division by 2 for odd integers in T, with a few other discrepancies in the upper parts that are still unclear to me). The above functions indeed verify
T = H^{-1} \circ U \circ H
by using the conjugacy map
H(n) = (a-2)n+b
which is not obvious (at first glimpse). One has to perform careful calculations. Somehow, it’s a deep relationship between two distinct mathematical objects. In fact, I did research on this topic some time ago, so this led me to go back into it.
The case a=3 and b=1 is already well documented, but I’m not aware of a particular paper with the general formulation of the conjugacy map H. Maybe @Failix has a matricial formalism for it in terms of tricot.