I am interested in finding homothetic transformations in the Poincaré upper half plane. I heard that unlike $\mathbb{R}^n$ we don't have an homothetic transform for every $\lambda \in \mathbb{R}^+$.
So I have now two questions. Do homothetic transforms have a general expression? And what are the extremal possible ratios for homothetic transforms?
Thanks, Emmanuel
A homothetic transformation of the Euclidean plane preserves the absolute values of angles but scales areas by $\lambda^2$. In hyperbolic geometry, the area of a polygon is equal to the its angle deficit. So as long as its angles don't change, its area can't change either. For this reason, the identity transformation with $\lambda=1$ is the only homothetic transformation matching your description. One might perhaps argue for $\lambda=0$ (maps the whole plane to a single point) or $\lambda=-1$ (reflection in a point) as valid alternatives, but since you restricted the setup to $\lambda\in\mathbb R^+$, those cases should be excluded.