On the Absence of Irrational Eigenvalues in Multidimensional Cellular Automata

Cellular automata are discrete dynamical systems in which complex global behaviour emerges from simple local interactions. While their dynamical and spectral properties are relatively well understood in one dimension, the multidimensional case remains largely unexplored. In this work, we investigate the spectral structure of cellular automata in arbitrary dimension from both topological and measure-theoretic perspectives. We first show that any multidimensional cellular automaton admitting fully-blocking patterns necessarily possesses a non-trivial periodic factor. We then provide a topological description of the spectrum that emphasizes the role of periodic and quasi-periodic components in the global dynamics. From a measure-theoretic viewpoint, we prove that for surjective cellular automata, the presence of eventual temporal periodicity in the trace of almost every configuration on a fixed finite window excludes the existence of irrational measurable eigenvalues. Moreover, we establish a rigidity result linking equicontinuity to spectral behaviour: if the set of equicontinuity points has positive measure, then either a rational measurable eigenvalue exists or every measurable eigenfunction vanishes on that set. As applications, we recover a purely rational spectrum for equicontinuous cellular automata and derive strong spectral restrictions for automata admitting fully blocking patterns. These results extend several classical one-dimensional spectral properties to higher dimensions and contribute to a deeper understanding of order and regularity in multidimensional cellular automata.