Piecewise Linear Hamiltonian Flows Associated to Zero-Sum Games: Transition Combinatorics and Questions on Ergodicity

TL;DR

This paper studies a class of piecewise affine Hamiltonian systems with straight-line orbits, providing initial classification and revealing complex orbit structures that raise questions about their ergodic properties.

Contribution

It offers the first classification of these systems and uncovers the rich, intricate orbit-structure of their flows.

Findings

The orbit-structure of these Hamiltonian flows is surprisingly complex.

A classification scheme for these piecewise affine Hamiltonian systems is proposed.

The systems exhibit rich combinatorial properties related to transition dynamics.

Abstract

In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbit-structure of the flow of such a differential equation is surprisingly rich.