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Uniqueness of aperiodic kneading sequences

Author: K. M. Brucks
Journal: Proc. Amer. Math. Soc. 107 (1989), 223-229
MSC: Primary 58F08; Secondary 26A18, 54H20
MathSciNet review: 979221
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Abstract: The trapezoidal function $ {f_e}(x)$ is defined for fixed $ e \in (0,1/2)$ by $ {f_e}(x) = (1/e)x$ for $ x \in [0,e],{f_e}(x) = 1$ for $ x \in (e,1 - e)$, and $ {f_e}(x) = (1/e)(1 - x)$ for $ x \in [1 - e,1]$. For a given $ e$ and the associated one-parameter family of maps $ \{ \lambda {f_e}(x)\vert\lambda \in [0,1]\} $, we show that if $ A$ is an aperiodic kneading sequence, then there is a unique $ \lambda \in [0,1]$ so that the itinerary of $ \lambda $ under the map $ \lambda {f_e}$ is $ A$. From this, we conclude that the "stable windows" are dense in $ [0,1]$ for the one-parameter family $ \lambda {f_e}$.

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Article copyright: © Copyright 1989 American Mathematical Society

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