7 - Dynamics in dimension zero

Summary

Zero-dimensional dynamical systems

Due to the existence of arbitrarily fine covers which are at the same time partitions (by disjoint open sets – we call them “clopen covers”), zero-dimensional dynamical systems allow us to switch between the topological and measuretheoretic dynamical notions easier than any other systems.

In order to understand zero-dimensional dynamical systems it suffices to understand subshifts and their countable joinings.

Definition 7.1.1 By a subshift we mean a topological dynamical system (X, T, S), where X is a closed, shift-invariant subset of ∧S′, ∧ is a finite set (called alphabet), and T denotes the shift map σ restricted to X. Both S′ and S stand for either ℕ₀ or ℤ but we require that S ⊂ S′. Shift-invariance of X is understood as σ(X) ⊂ X or σ(X) = X, depending on whether S = ℕ₀ or ℤ, respectively.

Note that S′ = ℤ implies that T is injective.

Definition 7.1.2 By a symbolic array system we will mean a topological dynamical system (X, T, S), where X consists of symbolic arrays of the form x = (x_k,n)_{k∈ℕ,n∈S′} (S ⊂ S′ like for subshifts), where for each k, n, x_k,n belongs to a finite set ∧_k not depending on n or x ∈ X, called the alphabet in row k. The transformation T is the restriction to X of the left shift map on arrays

The above definition implicitly requires that X be closed and invariant under T (recall that the meaning of invariance depends on S).