Geometry and Dynamics of the Besicovitch and Weyl Spaces

TL;DR

This paper explores the geometric and dynamical properties of Cantor subshifts within Besicovitch and Weyl spaces, revealing their homotopy classifications, automata behaviors, and the nonexistence of certain transitive cellular automata.

Contribution

It provides a comprehensive analysis of the geometric structure of subshifts and characterizes cellular automata behaviors in these spaces, including new proofs and classifications.

Findings

Sofic shifts correspond exactly to homotopy classes of simplicial complexes.

Characterization of contracting and isometric cellular automata in Besicovitch and Weyl spaces.

Proof that no transitive cellular automata exist in the Besicovitch space.

Abstract

We study the geometric properties of Cantor subshifts in the Besicovitch space, proving that sofic shifts occupy exactly the homotopy classes of simplicial complexes. In addition, we study canonical projections into subshifts, characterize the cellular automata that are contracting or isometric in the Besicovitch or Weyl spaces, study continuous functions that locally look like cellular automata, and present a new proof for the nonexistence of transitive cellular automata in the Besicovitch space.