Abstract
We define rectangle exchange transformations analogously to interval exchange transformations. An interval exchange transformation is a mapping of the unit interval onto itself obtained by cutting the interval up into a finite number of subintervals and rearranging the pieces. A rectangle exchange transformation is a mapping of the unit square onto itself obtained by cutting the square up into a finite number of rectangular pieces and rearranging the pieces. We give a minimality condition for rectangle exchange transformations. We deal with various examples of ergodic rectangle exchange transformations. Related questions are discussed.
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References
Cassels, J. W.: An Introduction to Diophantine Approximation. Cambridge: University Press. 1957.
Conze, J.-P.: Equirépartition et ergodicité de transformations cylindriques. Séminaire de Probabilités. Rennes: 1976.
Kakutani, S.: Examples of ergodic measure preserving transformations which are weakly mixing but not strongly mixing. In: Recent Advances in Topological Dynamics, pp. 143–149. Ed. byA. Beck. Lect. Notes Math. 318. Berlin-Heidelberg-New York: Springer. 1973.
Keane, M.: Interval exchange transformations. Math. Z.141, 25–31 (1975).
Keane, M.: Non-ergodic interval exchange transformations. Isr. J. Math.26, 188–196 (1977).
Keane, M., Rauzy, G.: Stricte ergodicité des échanges d'intervalles. Preprint.
Khintchine, A.: Kettenbrüche. Leipzig: Teubner. 1956.
Veech, W. A.: Interval exchange transformations. Journal d'Analyse Mathématique33, 222–272 (1978).
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Haller, H. Rectangle exchange transformations. Monatshefte für Mathematik 91, 215–232 (1981). https://doi.org/10.1007/BF01301789
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DOI: https://doi.org/10.1007/BF01301789