Observation Schemes and Irregularity in Linear Dynamics

Abstract

We develop a structural framework for irregularity in linear dynamics centered on the space $Υ$ 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 Cesàro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(μ_m)$-irregular and distributionally $(μ_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $Υ$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.