(G,\mu)- Quadratic Stochastic Operators

TL;DR

This paper introduces a new class of quadratic stochastic operators on finite Abelian groups, analyzing their convergence, periodicity, and stability properties using the concept of s(d)-invariant subgroups.

Contribution

It defines (d,bc)-quadratic stochastic operators and characterizes their long-term behavior, including convergence to fixed points and periodic trajectories.

Findings

Almost all trajectories converge to the simplex center.

Periodic trajectories are explicitly characterized.

Conditions for regularity and periodicity are established.

Abstract

We consider a new subclass of quadratic stochastic (evolutionary) operators on the simplex indexed by a finite Abelian group G with heredity law \mu. With the help of the notion of s(\mu)-invariant subgroups, where s(\mu) denotes the support of \mu in G, we prove that almost all (w.r.t.\ Lebesgue measure) trajectories of such operators converge to a unique fixed point which is the center of the simplex. We also identify and describe the periodic trajectories of the operator and give conditions for regularity and periodicity.