In his response to my question Conjugating the Lyness 5-cycle into a rotation of the plane, Francois Brunault provided an explicit conjugacy between the Lyness order-5 map and a 72-degree rotation, viewed as birational maps of projective 2-space.
However, Brunault’s map $g$ is not subtraction-free, so it does not tropicalize. So it is not clear to me whether the order-5 map of the plane to itself that sends $(x,y)$ to $(y,z)$ with $z = \max(y,0) - x$ is conjugate to a 72-degree rotation of the plane via a continuous piecewise linear conjugacy. Perhaps this is the same as asking whether the associated maps of tropical projective 2-space are conjugate via an automorphism of tropical projective 2-space.
