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Interval translation mappings. (English) Zbl 0836.58026
Let \(\{I\}\) denote the unit interval \(\{I\}= [0, a)\), \(a> 0\). Given an interval translation map \(T: \{I\}\to \{I\}\) the authors show that under certain arithmetical conditions the invariant set \(\Omega(T)= \bigcap_{n\geq 0} T^n(\{I\})\) is a finite union of intervals and the restriction of \(T\) to this set is an interval exchange transformation. The authors then construct an example of an interval translation map that translates just three intervals but \(\Omega(T)\) is a Cantor set of fractional Hausdorff dimension. The paper concludes with descriptions of many open problems regarding these maps.

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
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