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A case of monotone ratio growth for quadratic-like mappings

Pałuba W.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

381-393

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 fa = -|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 ..., or-in the quadratic family-to the parameters c belongs to [-1,0] in the Mandelbrot set. We allow the exponent alpha to be an arbitrary real number greater than 1.