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 | References | Similar Articles | Additional Information

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 -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

Email:
jerome.buzzi@u-bourgogne.fr

DOI:
https://doi.org/10.1090/S0002-9947-97-01873-4

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