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

Robert L. Devaney

Displaying similar documents to “Structure of the McMullen domain in the parameter planes for rational maps”

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

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Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let $M ₙ^{n + 1}$ , n ∈ ℕ, be the n-dimensional Menger continuum in $ℝ^{n + 1}$ , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $M ₙ^{n + 1}$ . We consider the topological group...

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Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s ≥ 3. Also let $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

When $C_{p} (X)$ is domain representable

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Let M be a metrizable group. Let G be a dense subgroup of $M^{X}$ . We prove that if G is domain representable, then $G = M^{X}$ . The following corollaries answer open questions. If X is completely regular and $C_{p} (X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_{p} (X,)$ is subcompact, then X is discrete.

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Let X be a Riemann domain over $ℂ^{k} \times ℂ^{ℓ}$ . If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P \subset ℂ^{k}$ such that every slice $X_{a}$ of X with a∉ P is a region of holomorphy with respect to the family ${f |}_{X_{a}} : f \in ℱ$ .

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Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^{P} ≅ Q^{Q}$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P ≅ Q$ .

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Consider the family uₜ = Δu + G(u), t > 0, $x \in Ω_{ε}$ , $\partial_{ν_{ε}} u = 0$ , t > 0, $x \in \partial Ω_{ε}$ , $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$ , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$ . If G is dissipative, then equation $(E_{ε})$ has a global attractor $_{ε}$ . We identify a “limit” equation for the family $(E_{ε})$ , prove convergence of trajectories and establish an upper semicontinuity result for the family $_{ε}$ as ε → 0⁺. ...

A domain whose envelope of holomorphy is not a domain

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We construct a domain of holomorphy in $ℂ^{N}$ , N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in $ℂ^{N}$ .

Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato, Eiichi Matsuhashi (2006)

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C (X, I^{p + 2 k + 1 - i})$ such that the diagonal product $f \times g : X \to Y \times I^{p + 2 k + 1 - i}$ is an (i+1)-to-1 map is a dense $G_{δ}$ -subset of $C (X, I^{p + 2 k + 1 - i})$ . In this paper, we prove that if f: X → Y is as above and $D_{j}$ (j = 1,..., k) are superdendrites, then the set of maps h in $C (X, \prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ such that $f \times h : X \to Y \times (\prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ is (i+1)-to-1 is a dense $G_{δ}$ -subset of $C (X, \prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ for each 0 ≤ i ≤ p.

On ramified covers of the projective plane II: Generalizing Segre’s theory

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

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In this note, we prove that there is no transcendental entire function $f (z) \in ℚ [[z]]$ such that $f (ℚ) \subseteq ℚ$ and $den f (p / q) = F (q)$ , for all sufficiently large $q$ , where $F (z) \in ℤ [z]$ .

3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

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The 3x+k function $T_{k} (n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_{k} (\cdot)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_{k} (\cdot)$ . We consider the generating functions $f_{k, m} (z) = \sum_{n > 0, n \to m} z^{n}$ , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k, m} (z)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold...

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We prove that the Niemytzki plane is $ϰ$ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a $ϰ$ -metrizable space. Also, we give a characterisation of $ϰ$ -metrizable spaces which is modelled on the version described by Chigogidze.

An irrational problem

Franklin D. Tall (2002)

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by $U \cap M : U \in \cap M$ . Suppose $X_{M}$ is homeomorphic to the irrationals; must $X = X_{M}$ ? We have partial results. We also answer a question of Gruenhage by showing that if $X_{M}$ is homeomorphic to the “Long Cantor Set”, then $X = X_{M}$ .

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials $P_{c} (z) = z^{d} + c$ for c ∈ ℂ. Denote by $J_{c}$ the Julia set of $P_{c}$ and let $ℳ_{d} = c | J_{c} i s c o n n e c t e d$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c ₀ \in \partial ℳ_{d}$ : those for which the critical point 0 is not recurrent by $P_{c ₀}$ and without parabolic cycles. The Hausdorff dimension of $J_{c}$ , denoted by $H D (J_{c})$ , does not depend continuously on c at such $c ₀ \in \partial ℳ_{d}$ ; on the other hand the function $c \mapsto H D (J_{c})$ is analytic in $ℂ - ℳ_{d}$ . Our first result asserts that there is still some...

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Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

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Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...