Period structure for pointwise periodic isometries of continua. (English) Zbl 0674.54020

Iwanik, Anzelm

Acta Univ. Carol., Math. Phys. 29, No. 2, 19-21 (1988).

An isometry T of a compact metric space X is said to be pointwise periodic if for every \(x\in X\) there exists \(n\in N\) such that \(T^ nx=x\). A subset S of N is said to be (a) finitely based if it has finitely many minimal elements when partially ordered by the divisibility relation \(m| n\); (b) connected if it forms a connected subgraph of the graph of the partially ordered set (N,\(|)\). Let S(X,T) stand for the set of all minimal periods of points of X. The following result is shown. If \(\emptyset \neq S\subset N\) is finitely based and connected, then there exists a continuum X and a pointwise periodic isometry T such that \(S=S(X,T)\). Conversely, if T is a pointwise periodic isometry of a continuum X then S(X,T) is finitely based and connected.

Reviewer: J.J.Charatonik

MSC:

54E40	Special maps on metric spaces
54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54F15	Continua and generalizations

Keywords:

finitely based isometry; pointwise periodic isometry