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1

From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

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Barański K.

Fundamenta Mathematicae

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2001

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tom 168

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nr 1

1-55

EN

This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a fixed point of multiplier 1. Then it bifurcates into two paths of Mandelbrot-like sets, contained respectively in the set of maps with exotic or non-exotic basins. The non-exotic path ends at a Mandelbrot-like set in cubic polynomials.

2

From Newton’s method to exotic basins Part I: The parameter space

100%

Barański K.

Fundamenta Mathematicae

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1998

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tom 158

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nr 3

249-288

EN

This is the first part of the work studying the family $\mathfrak{F}$ of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of $\mathfrak{F}$ and give a detailed study of the subfamily $ℱ_2$ consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in $ℱ_2$ from Newton maps to maps with so-called exotic basins.

3

Hausdorff dimension and measures on Julia sets of some meromorphic maps

100%

Barański K.

Fundamenta Mathematicae

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1995

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tom 147

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nr 3

239-260

EN

We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f(z) = h(exp \frac{2πi}{T}z)$ where h is a rational function or, equivalently, the maps $˜f(z) = exp (\frac{2πi}{h}(z))$. When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1. From this we obtain some lower bound for HD(J(f)) and real-analyticity of HD(J(f)) with respect to f. As an example the family $f_λ(z)=λ tan z$ is studied. We estimate $HD(J(f_λ))$ near λ = 0 and show it is a monotone function of real λ.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Barański K.

1 2001

1 1998

1 1995

From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

From Newton’s method to exotic basins Part I: The parameter space

Hausdorff dimension and measures on Julia sets of some meromorphic maps