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1
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Structure of the McMullen domain in the parameter planes for rational maps

100%
Devaney R.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 3
267-285
EN
We show that, for the family of functions $F_{λ}(z) = zⁿ + λ/zⁿ$ where n ≥ 3 and λ ∈ ℂ, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of $F_{λ}$ is homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.
2
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Intertwined internal rays in Julia sets of rational maps

100%
Devaney R.
Fundamenta Mathematicae
|
2009
|
tom 206
|
nr 1
139-159
EN
We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.
3
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Dynamic classification of escape time Sierpiński curve Julia sets

64%
Devaney R., Pilgrim K.
Fundamenta Mathematicae
|
2009
|
tom 202
|
nr 2
181-198
EN
For n ≥ 2, the family of rational maps $F_{λ}(z) = zⁿ + λ/zⁿ$ contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ ≥ 3. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of n and κ.
4
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Hyperbolic components of the complex exponential family

51%
Devaney R., Fagella N., Jarque X.
Fundamenta Mathematicae
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2002
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tom 174
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nr 3
193-215
EN
We describe the structure of the hyperbolic components of the parameter plane of the complex exponential family, as started in [1]. More precisely, we label each component with a parameter plane kneading sequence, and we prove the existence of a hyperbolic component for any given such sequence. We also compare these sequences with the more commonly used dynamical kneading sequences.
5
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A dynamical invariant for Sierpiński cardioid Julia sets

45%
Blanchard P., Cuzzocreo D., Devaney R., Fitzgibbon E., Silvestri S.
Fundamenta Mathematicae
|
2014
|
tom 226
|
nr 3
253-277
EN
For the family of rational maps zⁿ + λ/zⁿ where n ≥ 3, it is known that there are infinitely many small copies of the Mandelbrot set that are buried in the parameter plane, i.e., they do not extend to the outer boundary of this set. For parameters lying in the main cardioids of these Mandelbrot sets, the corresponding Julia sets are always Sierpiński curves, and so they are all homeomorphic to one another. However, it is known that only those cardioids that are symmetrically located in the parameter plane have conjugate dynamics. We produce a dynamical invariant that explains why these maps have different dynamics.
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