Monomial maps in two dimensions (Q1432722)

Let \(f\) be a rational self-map of \(\mathbb{P}^2\). We note \(d_{\text{t}}\) the topological degree and \(\deg (f)\) the algebraic degree of \(f\). The last one is the integer given by the action of \(f^*\) on \(H^{1,1}(\mathbb{P}^2,\mathbb{R})\). The limit \(\lambda\) of the sequence \((\deg(f^n)^{1 \over n})_n\) exists and is called the first dynamical degree of \(f\). Note that the ergodic properties of \(f\) are closely related with the ratio \({d_{\text{t}} \over \lambda}\). When \(f\) is algebraically stable (AS), the sequence \(\deg(f^n)_n\) is multiplicative, and this ratio is \({d_{\text{t}} \over \deg(f)}\). We recall that \(f\) is not AS when there exists a curve which is contracted to an indeterminacy point of \(f^n\), for some \(n \geq 1\). Even if a map is not AS, it is sometimes possible to find a birational change of coordinates \(\varphi : X \to \mathbb{P}^2\) such that the lifted map \(\tilde f : X \to X\) becomes AS. This is the case for the (more general) class of birational maps of complex surfaces [cf. the article of \textit{J. Diller} and \textit{C. Favre}, Am. J. Math. 123, 1135--1169 (2001; Zbl 1112.37308)]. In the article under review, we find examples of (noninvertible) rational maps of \(\mathbb{P}^2\) for which there does not exist such a change of coordinates. The author considers the family given by the monomial maps \((z,w) \mapsto (z^a w^b,z^c w^d)\) and describes completely the situation. More precisely, he gives necessary and sufficient conditions depending on the spectral properties of the matrix \([a,b,c,d]\) for which \(f\) becomes AS after a birational change of coordinates. There are special cases for which this is not possible. Note that the proof relies on elementary properties of toric geometry.