A classification of two dimensional integrable mappings and rational elliptic surfaces

TL;DR

This paper classifies two-dimensional integrable mappings by analyzing their actions on rational elliptic surfaces, revealing new classes of mappings that exchange fibers and conditions for certain surface types.

Contribution

It introduces a classification framework for integrable mappings based on their fiber actions and characterizes when generalized Halphen surfaces become Halphen surfaces of index m.

Findings

Identification of classes of integrable mappings exchanging fibers

Conditions for generalized Halphen surfaces to become Halphen surfaces of index m

Extension of QRT mappings classification

Abstract

We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which exchange fibers. We also show an equivalent condition when a generalized Halphen surface becomes a Halphen surface of index m.