On products of skew rotations

TL;DR

This paper investigates the asymptotic behavior of trajectories resulting from the composition of two Hamiltonian-induced shifts in the plane, with applications to population genetics models.

Contribution

It analyzes the asymptotic properties of plane transformations formed by products of Hamiltonian shifts, extending understanding of their long-term dynamics.

Findings

Characterization of trajectory asymptotics for product transformations

Conditions under which trajectories exhibit specific long-term behaviors

Insights applicable to models in population genetics

Abstract

Let $H_1(p,q)$, $H_2(p,q)$ be two time-independent Hamiltonians with one degree of freedom and $\{S_1^t\}$, $\{S_2^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by $H_1$, $H_2$. In some problems of population genetics there appear the transformations of the plane having the form $T^{(h_1,h_2)}=S^{h_2}_2\cdot S_1^{h_1}$ under some conditions on $H_1$, $H_2$. We study in this paper asymptotical properties of trajectories of $T^{(h_1,h_2)}$.