Algebraic entropy for algebraic maps

TL;DR

This paper extends the concept of algebraic entropy from rational maps to certain algebraic maps, providing a measure of their complexity with examples demonstrating positive entropy and applications to Backlund transformations.

Contribution

It introduces a new definition of algebraic entropy for algebraic maps, preserving key properties from the rational case and broadening the scope of entropy analysis.

Findings

Defined algebraic entropy for algebraic maps.

Provided examples with positive entropy.

Applied to Backlund transformations.

Abstract

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.