A note on the space of evolutionary operators in population genetics and folding dynamics

TL;DR

This paper characterizes the space of polynomial maps on the simplex relevant to population genetics, introduces folding maps as boundary cases, and explores their dynamical properties using numerical methods.

Contribution

It provides a rigorous description of the space of polynomial maps, introduces folding maps as a new class, and investigates their ergodic and mixing behaviors.

Findings

The space of bounded-degree polynomial maps is compact and convex.

Folding maps generalize logistic maps to higher dimensions and degrees.

Numerical analysis reveals diverse ergodic and mixing properties near folding maps.

Abstract

Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional progress has been made in finding particular solutions to these systems, our knowledge of the general properties of the space of all possible dynamical systems of this kind is still limited. We prove that the space of bounded-degree polynomial maps of the unit simplex to itself is a compact and convex subset of a Euclidean space. We provide an explicit characterization of such a space and of its boundary. A special class of maps in the boundary, the folding maps, which generalize the logistic map for any dimension and degree are defined and constructed. Finally, we use numerical methods to study the ergodic and mixing properties of maps in the neighborhood of several of these folding maps.