A Case of Monotone Ratio Growth for Quadratic-Like Mappings
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 381-393
MSC: Primary 37D05.
DOI: 10.4064/ba52-4-4
Abstract
This is a study of the monotone (in parameter) behavior of the ratios of the consecutive intervals in the nested family of intervals delimited by the itinerary of a critical point. We consider a one-parameter power-law family of mappings of the form $f_a=-|x|^{\alpha}+a$. Here we treat the dynamically simplest situation, before the critical point itself becomes strongly attracting; this corresponds to the kneading sequence $RRR\ldots, $ or—in the quadratic family—to the parameters $c\in [-1,0]$ in the Mandelbrot set. We allow the exponent $\alpha$ to be an arbitrary real number greater than 1.
