# Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlev\'e equations

## TL;DR

This paper investigates how the choice of elliptic fiber affects the deautonomization of integrable 2D mappings into discrete Painlevé equations, revealing new equations and geometric reconstruction techniques.

## Contribution

It introduces a fiber-dependent deautonomization method for integrable mappings and develops a geometric reconstruction technique for deriving explicit Painlevé equations.

## Key findings

- Different fibers yield various discrete Painlevé equations.
- A new technique for reconstructing mappings from geometric data.
- Discovery of new elliptic Painlevé equations with novel symmetry groups.

## Abstract

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlev\'e equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlev\'e equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlev\'e equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlev\'e equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.