Algebraic aspects of symbolic dynamics (Q2711282)

These lectures are a short and friendly introduction to the recent important developments in the classification of subshifts of finite type up to topological conjugacy. NEWLINENEWLINENEWLINEStarting with a quick overview of some natural classes of mixing subshifts (finite type, `almost' finite type, sofic, shifts with specification, synchronized shifts, and finally the coded shifts), the author goes on to sketch the pervasive connections between shifts of finite type (viewed as edge shifts) and properties of the associated adjacency matrix by expanding on a `dictionary' relating dynamical properties to matrix properties. Then isomorphism and eventual isomorphism are explained, along with their associated notions of strong shift equivalence and shift equivalence for matrices, and flow equivalence is described. NEWLINENEWLINENEWLINEWith this background in place, the author goes on to describe some of the powerful ideas from ordered algebra that have been used to attack the problem of deciding when two shifts of finite type are isomorphic, and to study the group of automorphisms of a shift of finite type. In these last sections little is proved, but the recent exciting progress on how sign-gyration conditions and the Factorization Theorem of \textit{K. H. Kim}, \textit{F. W. Roush} and \textit{J. B. Wagoner} [J. Am. Math. Soc. 5, 191-212 (1992; Zbl 0749.54012)] have resulted in a proof that shift equivalence does not force strong shift equivalence even for irreducible matrices is described and motivated. NEWLINENEWLINENEWLINEThese notes are an excellent introduction to the new directions in symbolic dynamics opened up by this result.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00028].