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On the classes of Lipschitz and smooth conjugacies of unimodal maps

100%
Pałuba W.
Fundamenta Mathematicae
|
2004
|
tom 183
|
nr 3
215-227
EN
Under very mild assumptions, any Lipschitz continuous conjugacy between the closures of the postcritical sets of two C¹-unimodal maps has a derivative at the critical point, and also on a dense set of its preimages. In a more restrictive situation of infinitely renormalizable maps of bounded combinatorial type the Lipschitz condition automatically implies the C¹-smoothness of the conjugacy. Here the critical degree can be any real number α > 1.
2
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A Case of Monotone Ratio Growth for Quadratic-Like Mappings

100%
Pałuba W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
tom 52
|
nr 4
381-393
EN
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 $f_a = -|x|^{α} + a$. 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 ∈ [-1,0] in the Mandelbrot set. We allow the exponent α to be an arbitrary real number greater than 1.
3
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The Lipschitz condition for the conjugacies of Feigenbaum-like mappings

51%
Pałuba W.
Fundamenta Mathematicae
|
1989
|
tom 132
|
nr 3
227-258
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