Non-uniform ergodic properties of Hamiltonian flows with impacts

TL;DR

This paper investigates the ergodic behavior of Hamiltonian flows with impacts in star-shaped polygons, revealing how energy levels influence topology, ergodicity, and the coexistence of periodic and ergodic components.

Contribution

It provides a detailed analysis of how impact boundaries and oscillator harmonicity affect the ergodic properties and topology of Hamiltonian flows in polygonal domains.

Findings

Flow on level sets is conjugated to translation flows on translation surfaces.

Unique ergodicity holds for almost all partial energies when oscillators are non-resonant or non-harmonic.

Existence of energy intervals with coexisting periodic ribbons and ergodic components.

Abstract

The ergodic properties of two uncoupled oscillators, a horizontal and vertical one, residing in a class of non rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly larger than one. When at least one of the oscillators is un-harmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is unique ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components co-exist. We prove that for almost all partial energies in such segments the motion is unique ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.