Specification on the interval

Buzzi, Jérôme

doi:10.1090/S0002-9947-97-01873-4

Specification on the interval

Author: Jérôme Buzzi
Journal: Trans. Amer. Math. Soc. 349 (1997), 2737-2754
MSC (1991): Primary 58F03; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-97-01873-4
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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