Multi-dimensional symbolic dynamical systems

The purpose of this note is to point out some of the phenomena which arise in the transition from classical shifts of finite type X ⊂ A ℤ to multi-dimensional shifts of finite type X ⊂ A ℤ d, d ≥ 2, where A is a finite alphabet. We discuss rigidity properties of certain multi-dimensional shifts, such as the appearance of an unexpected intrinsic algebraic structure or the scarcity of isomorphisms and invariant measures. The final section concentrates on group shifts with finite or uncountable alphabets, and with the symbolic representation of such shifts in the latter case.

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