Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

Abstract

In this paper we address the problem of computing $\text{deg}(f^n)$, the degrees of iterates of a birational map $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$. For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of $f$, based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by $f$, and the propagation of these indices along orbits of $f$. For maps admitting algebraically stable modifications $f_X:X\to X$, where $X$ is a variety obtained from ${\mathbb{P}}^N$ by a finite number of blow-ups, this method leads to an algorithm producing a finite system of recurrent equations relating the degrees and indices of iterated pull-backs of linear polynomials. We illustrate the method by three representative two-dimensional examples. It is actually applicable in any dimension, and we will provide a number of three-dimensional examples as a separate companion paper.

55 pages, one figure