Uniqueness of aperiodic kneading sequences

Abstract

The trapezoidal function f e ( x ) {f_e}(x) is defined for fixed e ∈ ( 0 , 1 / 2 ) e \in (0,1/2) by f e ( x ) = ( 1 / e ) x {f_e}(x) = (1/e)x for x ∈ [ 0 , e ] , f e ( x ) = 1 x \in [0,e],{f_e}(x) = 1 for x ∈ ( e , 1 − e ) x \in (e,1 - e) , and f e ( x ) = ( 1 / e ) ( 1 − x ) {f_e}(x) = (1/e)(1 - x) for x ∈ [ 1 − e , 1 ] x \in [1 - e,1] . For a given e e and the associated one-parameter family of maps { λ f e ( x ) | λ ∈ [ 0 , 1 ] } \{ \lambda {f_e}(x)|\lambda \in [0,1]\} , we show that if A A is an aperiodic kneading sequence, then there is a unique λ ∈ [ 0 , 1 ] \lambda \in [0,1] so that the itinerary of λ \lambda under the map λ f e \lambda {f_e} is A A . From this, we conclude that the "stable windows" are dense in [ 0 , 1 ] [0,1] for the one-parameter family λ f e \lambda {f_e} .