The branch of the theory of dynamical systems devoted to the investigation of the dynamical properties of shift maps \(\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}\) is known as *symbolic dynamics*. The most studied case is when the symbol space *K* is a finite or countably infinite discrete space (see for example [59, 70]).

Theorem 7.2.1 is in [44, 74]. Theorem 7.3.5 is a particular case of Proposition 1.9.A in [44].

As mentioned in the Notes on Chap. 6, Boltyanskiǐ [15, 16] gave examples of compact metrizable spaces *K* satisfying \({{\mathrm{stabdim}}}(K) < \dim (K)\). For such spaces *K*, we have that \({{\mathrm{mdim}}}(K^\mathbb {Z},\sigma ) < \dim (K)\) by Theorem 7.1.3. It would be interesting to find an example of a compact metrizable space *K* for which \({{\mathrm{mdim}}}(K^\mathbb {Z},\sigma ) < {{\mathrm{stabdim}}}(K)\).