Measures and Trajectory Properties in Oscillator Systems
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| Format: | Preprint |
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2025
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| _version_ | 1866915445695578112 |
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| author | Sakbaev, Vsevolod Volovich, Igor |
| author_facet | Sakbaev, Vsevolod Volovich, Igor |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measures and Trajectory Properties in Oscillator Systems Sakbaev, Vsevolod Volovich, Igor Dynamical Systems Mathematical Physics Functional Analysis Quantum Physics Primary 28D05, 37A05, 47A35, Secondary 37K10, 37N20 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. |
| title | Measures and Trajectory Properties in Oscillator Systems |
| topic | Dynamical Systems Mathematical Physics Functional Analysis Quantum Physics Primary 28D05, 37A05, 47A35, Secondary 37K10, 37N20 |
| url | https://arxiv.org/abs/2506.18093 |