6533b85efe1ef96bd12bfb3a
RESEARCH PRODUCT
Specification on the interval
Jérôme BuzziJérôme Buzzisubject
Discrete mathematicsProperty (philosophy)Lebesgue measureApplied MathematicsGeneral MathematicsSymbolic dynamicsPiecewiseMonotonic functionInterval (mathematics)Classification of discontinuitiesNatural classMathematicsdescription
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 β \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).
| year | journal | country | edition | language |
|---|---|---|---|---|
| 1997-01-01 | Transactions of the American Mathematical Society |