Studies on spaces of initial conditions for nonautonomous mappings of the plane

TL;DR

This paper investigates nonautonomous plane mappings using spaces of initial conditions, revealing that those with zero algebraic entropy and unbounded degree growth are classified as discrete Painlevé equations.

Contribution

It introduces the concept of spaces of initial conditions for nonautonomous systems and characterizes mappings with zero entropy as discrete Painlevé equations.

Findings

Mappings with zero algebraic entropy and unbounded degree growth are classified as discrete Painlevé equations.

Introduces the notion of spaces of initial conditions for nonautonomous systems.

Establishes properties of equations with spaces of initial conditions.

Abstract

We study nonautonomous mappings of the plane by means of spaces of initial conditions. First we introduce the notion of a space of initial conditions for nonautonomous systems and we study the basic properties of general equations that have spaces of initial conditions. Then, we consider the minimization of spaces of initial conditions for nonautonomous systems and we show that if a nonautonomous mapping of the plane with a space of initial conditions, and unbounded degree growth, has zero algebraic entropy, then it must be one of the discrete Painlev\'{e} equations in the Sakai classification.