Geodesic
A Case of Monotone Ratio Growth for Quadratic-Like Mappings
Waldemar Pałuba1
1 Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 4, pp. 381-393.

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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.
DOI : 10.4064/ba52-4-4
Keywords: study monotone parameter behavior ratios consecutive intervals nested family intervals delimited itinerary critical point consider one parameter power law family mappings form alpha here treat dynamically simplest situation before critical point itself becomes strongly attracting corresponds kneading sequence rrr ldots quadratic family parameters mandelbrot set allow exponent alpha arbitrary real number greater

Waldemar Pałuba 1

1 Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland 
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Waldemar Pałuba. A Case of Monotone Ratio Growth for Quadratic-Like Mappings. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 4, pp. 381-393. doi : 10.4064/ba52-4-4. http://geodesic.mathdoc.fr/articles/10.4064/ba52-4-4/

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