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

Krzysztof Barański

Displaying similar documents to “From Newton’s method to exotic basins Part I: The parameter space”

Computing Reidemeister classes

Davide Ferrario (1998)

Fundamenta Mathematicae

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In order to compute the Nielsen number N(f) of a self-map f: X → X, some Reidemeister classes in the fundamental group $π_{1} (X)$ need to be distinguished. In this paper some algebraic results are given which allow distinguishing Reidemeister classes and hence computing the Reidemeister number of some maps. Examples of computations are presented.

Linear orders and MA + ¬wKH

Zoran Spasojević (1995)

Fundamenta Mathematicae

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I prove that the statement that “every linear order of size $2^{ω}$ can be embedded in $(ω^{ω}, ≪)$ ” is consistent with MA + ¬ wKH.

The Arkhangel’skiĭ–Tall problem: a consistent counterexample

Gary Gruenhage, Piotr Koszmider (1996)

Fundamenta Mathematicae

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We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${[ω]}^{ω}$ , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

Categoricity of theories in Lκω , when κ is a measurable cardinal. Part 1

Saharon Shelah, Oren Kolman (1996)

Fundamenta Mathematicae

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We assume a theory T in the logic $L_{κ ω}$ is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.

Concordant sequences and integral-valued entire functions

Jonathan Pila, Fernando Rodriguez Villegas (1999)

Acta Arithmetica

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A classic theorem of Pólya shows that the function $2^{z}$ is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

W. Jurkat, D. Nonnenmacher (1994)

Fundamenta Mathematicae

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^{n}$ , we are led to an integration process over quite general sets $A \subseteq q ℝ^{n}$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

Self-homeomorphisms of the 2-sphere which fix pointwise a nonseparating continuum

Paul Fabel (1998)

Fundamenta Mathematicae

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We prove that the space of orientation preserving homeomorphisms of the 2-sphere which fix pointwise a nontrivial nonseparating continuum is a contractible absolute neighborhood retract homeomorphic to the separable Hilbert space $l_{2}$ .

Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

Alessandro Andretta, Alberto Marcone (1997)

Fundamenta Mathematicae

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We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $\prod_{2}^{0}$ -complete and that the set of Cauchy problems which locally have a unique solution is $\sum_{3}^{0}$ -complete. We prove that the set of Cauchy problems which have a global solution is...

The structure of atoms (hereditarily indecomposable continua)

R. Ball, J. Hagler, Yaki Sternfeld (1998)

Fundamenta Mathematicae

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Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $ϱ (x, y) = W (A_{x y})$ where $A_{x, y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has...

Homotopy orbits of free loop spaces

Marcel Bökstedt, Iver Ottosen (1999)

Fundamenta Mathematicae

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Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors $Ω_{λ} (R)$ and ℓ(R) together with algebra homomorphisms $e : Ω_{λ} (R) \to H * (Λ X)$ and $ψ : ℓ (R) \to H * (E S^{1} \times_{S^{1}} Λ X)$ . When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

Homeomorphisms of inverse limit spaces of one-dimensional maps

Marcy Barge, Beverly Diamond (1995)

Fundamenta Mathematicae

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We present a new technique for showing that inverse limit spaces of certain one-dimensional Markov maps are not homeomorphic. In particular, the inverse limit spaces for the three maps from the tent family having periodic kneading sequence of length five are not homeomorphic.

The Conley index for decompositions of isolated invariant sets

Andrzej Szymczak (1995)

Fundamenta Mathematicae

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A Lefschetz-type coincidence theorem

Peter Saveliev (1999)

Fundamenta Mathematicae

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A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: $I_{f g} = λ_{f g}$ , that is, the coincidence index is equal to the Lefschetz number. It follows that if $λ_{f g} \neq 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “point-like”...