zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI
[1] DOI: 10.1007/BF02790191 · Zbl 0602.28008 · doi:10.1007/BF02790191
[2] DOI: 10.1214/aop/1176995475 · Zbl 0377.28014 · doi:10.1214/aop/1176995475
[3] DOI: 10.1017/S0143385700006866 · Zbl 0756.58030 · doi:10.1017/S0143385700006866
[4] DOI: 10.1017/S0143385700004521 · Zbl 0657.28013 · doi:10.1017/S0143385700004521
[5] DOI: 10.1007/BF02790174 · Zbl 0455.28006 · doi:10.1007/BF02790174
[6] DOI: 10.1070/RM1967v022n05ABEH001224 · Zbl 0174.45501 · doi:10.1070/RM1967v022n05ABEH001224
[7] Queftelec, Substitution Dynamical Systems-Spectral Analysis, Springer Lecture Notes in Mathematics 1294. Springer (1980)
[8] Oseledets, Dokl. Acad. Sci. USSR 168 pp 1009– (1966)
[9] ManĂ©, Ergodic Theory and Differentiable Dynamics. Springer (1987) · doi:10.1007/978-3-642-70335-5
[10] Levitt, Prepublication du Laboratoire de Topologie et Geometric URA CNRS 1408 (1990)
[11] DOI: 10.1007/BF01236981 · Zbl 0278.28010 · doi:10.1007/BF01236981
[12] Katok, Sov. Math. Dokl. 14 pp 1104– (1973)
[13] DOI: 10.1007/BF02760527 · Zbl 0424.28004 · doi:10.1007/BF02760527
[14] Falconer, Fractal Geometry. John Wiley (1990)
[15] Cornfeld, Ergodic Theory. Springer (1982) · doi:10.1007/978-1-4615-6927-5
[16] Cassels, An Introduction to Diophantine Approximation (1957) · Zbl 0077.04801
[17] DOI: 10.1007/BF01244320 · Zbl 0839.28008 · doi:10.1007/BF01244320
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.