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

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
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