The Williams conjecture is false for irreducible subshifts. (English) Zbl 0989.37005

Kim, K. H.; Roush, F. W.

Ann. Math. (2) 149, No. 2, 545-558 (1999).

In this important paper the Williams conjecture is proved to be false. According to this conjecture, the shifts of finite type are classified (up to topological conjugacy) by shift equivalence. In [K. H. Kim and F. W. Roush, J. Am. Math. Soc. 5, 213-215 (1992; Zbl 0749.54013)] a counterexample in the case of reducible shifts of finite type was given. In the paper under review it is shown that the Williams conjecture is false also for irreducible shifts of finite type. The counterexample grows out of the factorization theorem of K. H. Kim, F. W. Roush and J. B. Wagoner [J. Am. Math. Soc. 5, 191-212 (1992; Zbl 0749.54012)] which states that the SGCC (sign-gyration-compatibility-condition) representation of the automorphism group of a shift of finite type factors through its dimension representation, and by certain explicit formulas. The proof of this theorem allowed the authors to define a certain relative cohomology class whose nontriviality gives a counterexample to Williams conjecture in the irreducible case. A simplification of the original proof of the factorization theorem is also given.

Reviewer: Ľubomír Snoha (Banská Bystrica)

MSC:

37B10	Symbolic dynamics
54H20	Topological dynamics (MSC2010)

Keywords:

sign gyration compatibility condition; Williams conjecture; topological conjugacy; shift equivalence; irreducible shifts of finite type; factorization theorem

Citations:

Zbl 0749.54013; Zbl 0749.54012