Uniqueness of Aperiodic Kneading Sequences

Abstract

Abstract.The trapezoidal function fe[x) is defined for fixed e 6 (0,1/2) by fe(x) = (l/e)x for x e [0,e], fe{x) =1 for x e (e, 1 -e), and fe(x) = (l/e)(l-jf) for x€ [l-e, 1], Foragiven e and the associated one-parameter family of maps {Xfe(x)\X e [0,1]} , we show that if A is an aperiodic kneading sequence, then there is a unique A 6 [0, 1] so that the itinerary of A under the map Xfe is A . From this, we conclude that the "stable windows" are dense in [0,1] for the one-parameter family Xfe .This note is mainly concerned with those maps which are trapezoidal. The trapezoidal function, fe, is defined for e G (0,1/2) by feix) = x/e for x G [0,e], feix) = l for xGie,l-e),and feix) = (1 -x)/e for xe [l-e,l]. For g unimodal and k G [0,1], the intinerary of k under the map kg, I 8ik), is referred to as the kneading sequence of kg [6]. BMS show that / sik) is shift maximal in the parity-lexicographical order when g is unimodal and k G [0,1 ] (throughout this note, when comparing kneading sequences the paritylexicographical order is used). Furthermore, if B is a finite or infinite shift maximal sequence, then there is some unimodal map g and some k G [0,1] so that I gik) = B . Thus any kneading sequence is shift maximal, and any shift maximal sequence is a kneading sequence. We note that if g is unimodal and