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

2023/03/28 by Jaume Alonso, Alonso, Jaume, Yuri B. Suris +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2303.15864

openalex publication_date 2023/03/28 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28

Abstract

In this paper we address the problem of computing deg(fⁿ), the degrees of iterates of a birational map f:ℙ^N→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→ 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.

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