Uniqueness of aperiodic kneading sequences

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 $\{ \lambda {f_e}(x)|\lambda \in [0,1]\}$, 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 $\lambda {f_e}$.