Zero Entropy for Some Birational Maps of C^2

TL;DR

This paper investigates specific birational maps of C^2 with zero entropy, analyzing their degree growth, invariant fibrations, and integrable cases, revealing diverse dynamical behaviors including periodicity and integrability.

Contribution

It characterizes zero entropy subfamilies of birational maps of C^2, providing explicit invariant fibrations and classifying their degree growth patterns.

Findings

Degree sequences grow periodically, linearly, quadratically, or exponentially.

Explicit invariant fibrations are constructed for zero entropy families.

All integrable and periodic mappings within the family are identified.

Abstract

This work deals with a special case of family of birational maps f : C2 -> C2 dynamically classified in [9]. In this work we study the zero entropy sub families of f. The sequence of degrees dn associated to the iterates of f is found to grow periodically, linearly, quadratically or exponentially. Explicit invariant fibrations for zero entropy families and all the integrable and periodic mappings inside the family f are given.