Irrational eigenvalues of almost expansive cellular automata

TL;DR

This paper proves that almost expansive cellular automata cannot have irrational eigenvalues, extending to surjective automata, thus clarifying spectral properties in cellular automata theory.

Contribution

It establishes that almost expansive and surjective cellular automata cannot possess irrational eigenvalues, completing a classification in spectral analysis.

Findings

Almost expansive cellular automata lack irrational eigenvalues.

Surjective cellular automata cannot have irrational eigenvalues for the uniform measure.

The result completes the spectral classification of cellular automata.

Abstract

We show that an almost expansive cellular automaton according to Gilman's classification cannot have irrational eigenvalues. This completes the proof that any surjective cellular automaton cannot have irrational eigenvalues for the uniform measure.