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arXiv · 2608.22616

Observation Schemes and Irregularity in Linear Dynamics

Manuel Saavedra
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Abstract

We develop a structural framework for irregularity in linear dynamics centered on the space $\Upsilon$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li--Yorke chaos, mean Li--Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Ces\`aro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(\mu_m)$-irregular and distributionally $(\mu_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $\Upsilon$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.

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Manuel Saavedra. 2026-08-23. Observation Schemes and Irregularity in Linear Dynamics. https://arxiv.org/abs/2608.22616

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