A hereditarily indecomposable tree-like continuum without the fixed point property. (English) Zbl 0954.54014
Answering a question of R. H. Bing [Am. Math. Mon. 76, 119-132 (1969; Zbl 0174.25902)], D. P. Bellamy has constructed [Houston J. Math. 6, 1-13 (1980; Zbl 0447.54039)] a tree-like continuum without the fixed point property and asked later in [Lect. Notes Pure Appl. Math. 170, 27-35 (1995; Zbl 0833.54023)] whether each hereditarily indecomposable continuum has the fixed point property. The same question was asked by B. Knaster in 1974 as Problem 69 of the Houston problem book [Lect. Notes Pure Appl. Math. 170, 365-398 (1995; Zbl 0828.54001)]. After 25 years the author answers the question of Knaster and Bellamy in the negative, not only by constructing the needed example, but by proving a much more general theorem which is the main result of the paper.
Theorem. For each positive integer \(j\) there exists a hereditarily indecomposable tree-like continuum \(X_j\) and a mapping \(h_j:X_j\to X_j\) such that \(h_j\) does not have periodic points of periods less than or equal to \(j\).
The paper consists of three sections. In the introduction a very informative historical and general background is presented. In the second section, using the technique introduced by the author and W. R. R. Transue in [Proc. Am. Math. Soc. 111, No. 4, 1165-1170 (1991; Zbl 0767.54034)], a mapping on a special continuum \(B\) is constructed whose inverse limit is hereditarily indecomposable. The mapping, together with constructions from author’s papers [Topology Appl. 46, No. 2, 99-106 (1992; Zbl 0770.54043); Trans. Am. Math. Soc. 348, No. 4, 1487-1519 (1996; Zbl 0863.54027)], are applied to prove the main result in section three. The References consist of 28 items.
Theorem. For each positive integer \(j\) there exists a hereditarily indecomposable tree-like continuum \(X_j\) and a mapping \(h_j:X_j\to X_j\) such that \(h_j\) does not have periodic points of periods less than or equal to \(j\).
The paper consists of three sections. In the introduction a very informative historical and general background is presented. In the second section, using the technique introduced by the author and W. R. R. Transue in [Proc. Am. Math. Soc. 111, No. 4, 1165-1170 (1991; Zbl 0767.54034)], a mapping on a special continuum \(B\) is constructed whose inverse limit is hereditarily indecomposable. The mapping, together with constructions from author’s papers [Topology Appl. 46, No. 2, 99-106 (1992; Zbl 0770.54043); Trans. Am. Math. Soc. 348, No. 4, 1487-1519 (1996; Zbl 0863.54027)], are applied to prove the main result in section three. The References consist of 28 items.
Reviewer: J.J.Charatonik (México)
MSC:
| 54F15 | Continua and generalizations |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
Citations:
Zbl 0174.25902; Zbl 0447.54039; Zbl 0833.54023; Zbl 0828.54001; Zbl 0767.54034; Zbl 0770.54043; Zbl 0863.54027References:
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