The cohomology of higher-dimensional shifts of finite type (Q1919868)

Combined review for the paper cited before and the present paper. The authors ask the question why `good' actions of the higher rank abelian groups \(\mathbb{Z}^d\) and \(\mathbb{R}^d\), for \(d\geq 2\) are much more rare and rigid than for similar actions on \(\mathbb{Z}\) and \(\mathbb{R}\). An example due to \textit{J. W. Kammeyer} [J. Anal. Math. 54, 113-163 (1990; Zbl 0701.28007) and Ergodic Theory Dyn. Syst. 12, No. 2, 267-282 (1992; Zbl 0761.28013)] where it was shown that every continuous cocycle for the shift action of \(\mathbb{Z}^d\) on the full \(d\)-dimensional \(k\)-shift with values in \(\mathbb{Z}/2\mathbb{Z}\) is continuously cohomologous to a homomorphism from \(\mathbb{Z}^d\) to \(\mathbb{Z}/2\mathbb{Z}\), illustrates this phenomenon. \textit{A. Katok} and \textit{R. J. Spatzier} [Publ. Math., Inst. Hautes Études Sci. 79, 131-156 (1994; Zbl 0819.58027)] have shown that every real valued Hölder 1-cocycle for an Anosov \(\mathbb{Z}^d\)-action on a compact manifold is Hölder cohomologous to a homomorphism. Mixing expansive actions by commuting toral automorphisms are examples of this. The first of these two papers is an extension of this result. Using the structure of \(\mathbb{Z}^d\)-actions by automorphisms of compact abelian groups, and techniques for proving the triviality of the first cohomology of higher rank abelian group actions, Katok and Schmidt prove that for \(d>1\), every real-valued Hölder cocycle of an expansive and mixing \(\mathbb{Z}^d\)-action by automorphisms of a compact, abelian group is Hölder cohomologous to a homomorphism. In the second paper, Schmidt looks at the cohomology of higher-dimensional shifts of finite type. Here the situation becomes more complicated. It is shown that if a \(d\)-dimensional shift of finite type \(X\) has a rich supply of homoclinic points (i.e., points which agree except on finitely many coordinates), and a certain specification property, then every Hölder cocycle for shift actions of \(\mathbb{Z}^d\) on \(X\) with values in a second countable group with a doubly invariant metric is Hölder cohomologous to a homomorphism. A number of examples including some well-known subshifts of finite type, and some zero entropy subshifts of finite type, are given.