Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

TL;DR

This paper extends a polynomial index method to compute the degrees of iterates of 3D birational maps, avoiding the need for algebraic stability, and demonstrates its effectiveness through several examples.

Contribution

It introduces a new polynomial index approach for 3D birational maps that simplifies degree calculations without requiring algebraic stability.

Findings

Method successfully computes degrees of iterates in 3D examples.

Approach avoids constructing algebraically stable models.

Demonstrates flexibility and applicability in complex cases.

Abstract

The goal of this paper is the exact computation of the degrees $\text{deg}(f^n)$ of the iterates of birational maps $f: \mathbb{P}^N \dashrightarrow \mathbb{P}^N$. In the preceding companion paper, a new method has been proposed based on the use of indices of polynomials associated to the local blow-ups used to resolve contractions of hypersurfaces by $f$, and on the control of the factorization of pull-backs of polynomials. This leads to recurrence relations for the degrees and the indices. We apply this method to several illustrative examples in three dimensions. These examples demonstrate the flexibility of the method which, in particular, does not require the construction of an algebraically stable lift of $f$, unlike the previously known methods based on the Picard group.