Strong shift equivalence of matrices in \(GL(2,\mathbb{Z})\). (English) Zbl 0768.58039

Williams, R. F.

Symbolic dynamics and its applications, Proc. AMS Conf. in honor of R. L. Adler, New Haven/CT (USA) 1991, Contemp. Math 135, 445-451 (1992).

[For the entire collection see Zbl 0755.00019.]
The main result of this work is the following Proposition. Given \(2 \times 2\) matrices \(A\), \(B\) over \(\mathbb{Z}^ +\) with \(\text{det }A = \pm 1\), the following are equivalent 1) \(A = RBR^{-1}\), for some \(R \in GL(2, )\), 2) \(A \sim_ s B\), 3) \(A \approx_ s B\), 4) there exist non- negative integral matrices \(R\), \(S\) such that \(A = RS\) and \(B = SR\).
Here \(\sim_ s\) and \(\approx_ s\) denote respectively the weak and the strong shift equivalences introduced by the author in an earlier work [Global Analysis, Proc. Sympos. Pure Math. 14, 341-361 (1970; Zbl 0213.504)].

Reviewer: C.S.Sharma (London)

MSC:

37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37E99	Low-dimensional dynamical systems
47A35	Ergodic theory of linear operators
54H20	Topological dynamics (MSC2010)

Keywords:

shift equivalence; symbolic dynamics; integral matrices

Citations:

Zbl 0213.504; Zbl 0755.00019