Monomial maps $\mathbb C^n \to \mathbb C^n$ are birational maps when they are based on a unimodular matrix. When $n=2$, their natural compactification exhibits a periodic behavior, reflecting the periodic behavior of continued fractions for quadratic irrationals.

The quotient of the compactification by the period map is one way to understand the desingularization of Hilbert modular surfaces at the cusps. This technique extends to higher dimensions, and gives insights into Hilbert modular forms.