Measures and Trajectory Properties in Oscillator Systems

TL;DR

This paper explores the complex trajectory behaviors in harmonic oscillator systems with different measures, revealing new phenomena in infinite-dimensional cases and establishing conditions for wandering and non-wandering points.

Contribution

It introduces new classes of trajectory behaviors in infinite-dimensional oscillator systems and provides conditions distinguishing between wandering and non-wandering points based on measure types.

Findings

Non-periodic trajectories in infinite-dimensional systems.

Non-wandering points on invariant tori in countable systems.

Conditions for absence of transitive trajectories in singular measure systems.

Abstract

This paper investigates the properties of trajectories in harmonic oscillator systems equipped with a point, absolutely continuous, or singular measure. As demonstrated in [30], infinite-dimensional linear flows of countable oscillator systems exhibit a new class of trajectory behavior. Specifically, these trajectories are non-periodic, and their projections onto any four-dimensional symplectic subspace fail to be dense in the corresponding projection of the invariant torus. Such trajectories do not arise in finite-dimensional systems, are non-generic for countable oscillator systems, but become generic in the continual case. We prove that for a countable harmonic oscillator system, every point on a non-degenerate invariant torus is a non-wandering point of the flow. In contrast, for a continual system with an absolutely continuous measure, all points on such a torus are wandering. Furthermore, for continual systems with a singular measure, we establish sufficient conditions on the measure and torus that rule out the existence of both transitive trajectories and non-wandering points. As an application, we exhibit a class of singular Bernoulli measures satisfying these conditions.