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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Specification on the interval

Author: Jérôme Buzzi
Journal: Trans. Amer. Math. Soc. 349 (1997), 2737-2754
MSC (1991): Primary 58F03; Secondary 54H20
MathSciNet review: 1407484
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Abstract: We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the $\beta$-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133–134) (for which we give a proof).

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Additional Information

Jérôme Buzzi
Affiliation: Université Paris-Sud, Bât. 425, 91405 Orsay, France
Address at time of publication: Université de Bourgogne, Lab. de Topologie, B.P. 400, 21011 Dijon Cedex, France

Keywords: One-dimensional dynamics, piecewise monotonic, piecewise continuous maps, specification property, Lebesgue measure in parameter space, symbolic dynamics, Hofbauer’s Markov diagram
Received by editor(s): September 12, 1995
Article copyright: © Copyright 1997 American Mathematical Society