Dynamics of \(\mathbb{Z}^d\) actions on Markov subgroups (Q2711283)

These notes are an introduction to multidimensional Markov shifts which have a group structure. They are called Markov subgroups. The alphabet of the Markov subgroup can be any compact group but there the author concentrates on the ones with a finite alphabet. Any expansive \(\mathbb{Z}^d\) action of a compact group can be represented as a Markov subgroup. The notes are concerned with the case where the group is totally disconnected but some general ideas are mentioned.NEWLINENEWLINENEWLINEFrom the introduction: The formulation of the problems considered and the results presented here can be found in the author's papers [Ergodic Theory Dyn. Syst. 7, 249-261 (1987; Zbl 0597.54039); with \textit{K. Schmidt}, Ergodic Theory Dyn. Syst. 9, 691-735 (1989; Zbl 0709.54023); in Dynamical systems, College Park, MD, 1986-87, Lect. Notes Math. 1342, 440-454 (1988; Zbl 0664.58029); in Symbolic dynamics and its applications, New Haven, CT, 1991, Contemp. Math. 135, 265-283 (1992; Zbl 0774.58021); Ergodic Theory Dyn. Syst. 13, No.~4, 705-735 (1993; Zbl 0799.58043)]. A comprehensive introduction to this subject can be found in [\textit{K. Schmidt}, Dynamical systems of algebraic origin, Prog. Math. 128, Birkhäuser, Basel (1995; Zbl 0833.28001)]. It contains much more than is presented here and thoroughly treats the case of Markov subgroups with a compact connected alphabet.NEWLINENEWLINENEWLINEThe lectures are organized as follows. Section 4.2 contains examples of one-dimensional Markov subgroups and then proves a structure theorem (Theorem 4.2.7) for one-dimensional Markov subgroups on a finite alphabet. Section 4.3 contains a discussion of some decidability questions for general two-dimensional Markov shifts and how they relate to one-dimensional Markov shifts, tilings and periodic points. Proposition 4.3.5 states that the general tiling problem is undecidable. This was proved by R. Berger and the notes do not contain a proof. Then the way these results relate to two-dimensional Markov subgroups is examined. The section concludes by observing that these decidability problems do not arise for Markov subgroups. In Section 4.4 an examination of two-dimensional Markov subgroups with \((\mathbb{Z}/2\mathbb{Z})\) as the alphabet is begun. The basic definitions are made and questions about entropy and directional dynamics are discussed. Section 4.5 contains the heart of these lectures. To each such Markov subgroup an ideal in the ring \((\mathbb{Z}/2\mathbb{Z})[x^{\pm 1},y^{\pm 1}]\) is associated. A systematic study of the relationship between the dynamical properties of the Markov subgroup and the algebraic properties of the ideal is begun. Section 4.6 contains a few remarks on the conjugacy and isomorphism questions about Markov subgroups studied in Section 4.5. Section 4.7 concludes the lectures by showing how the algebraic formulation of Section 4.5 can be extended to the study of all expansive \(\mathbb{Z}^d\) actions of compact groups.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00028].