Some interesting birational morphisms of smooth affine quadric $3$-folds

TL;DR

This paper investigates a family of birational maps on smooth affine quadric 3-folds, revealing their unique dynamical properties, including cohomological hyperbolicity and unusual growth rates of periodic points, through both theoretical and computational methods.

Contribution

It introduces and analyzes a new class of birational maps with distinctive dynamical behaviors on affine quadrics, combining rigorous proofs and numerical experiments.

Findings

Maps are cohomologically hyperbolic

Second dynamical degree is algebraic but not an algebraic integer

Logarithmic growth of periodic points is less than algebraic entropy

Abstract

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are cohomologically hyperbolic, b) their second dynamical degree is an algebraic number but not an algebraic integer, and c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on $\mathbb{C}^4$ preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.