Misiurewicz maps unfold generically (even if they are critically non-finite)

Sebastian van Strien

Displaying similar documents to “Misiurewicz maps unfold generically (even if they are critically non-finite)”

Cofinal $Σ_{1}^{1}$ and $Π_{1}^{1}$ subsets of $ω^{ω}$

Gabriel Debs, Jean Saint Raymond (1999)

Fundamenta Mathematicae

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We study properties of $\sum_{1}^{1}$ and $π_{1}^{1}$ subsets of $ω^{ω}$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement " $\forall α \in ω^{ω}, ω^{ω} \cap L (α)$ is bounded for ≤*".

Locally constant functions

Joan Hart, Kenneth Kunen (1996)

Fundamenta Mathematicae

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Let X be a compact Hausdorff space and M a metric space. $E_{0} (X, M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_{0} (X, M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_{0} (X, M)$ as a normed linear space. We also build three first countable...

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada, Michał Misiurewicz, L’ubomír Snoha (1999)

Fundamenta Mathematicae

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The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${(X_{i})}_{i = 1}^{\infty}$ and a sequence of continuous maps ${(f_{i})}_{i = 1}^{\infty}$ , $f_{i} : X_{i} \to X_{i + 1}$ , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_{n} ○ . . . ○ f_{2} ○ f_{1}$ . As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty}$ and...

Bimorphisms in pro-homotopy and proper homotopy

Jerzy Dydak, Francisco Ruiz del Portal (1999)

Fundamenta Mathematicae

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A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $t o w (H_{0})$ is an isomorphism if Y is movable. Recall that $(H_{0})$ is the full subcategory of $p r o - H_{0}$ ...

On absolute retracts of ω*

A. Bella, A. Błaszczyk, A. Szymański (1994)

Fundamenta Mathematicae

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An extremally disconnected space is called an absolute retract in the class of all extremally disconnected spaces if it is a retract of any extremally disconnected compact space in which it can be embedded. The Gleason spaces over dyadic spaces have this property. The main result of this paper says that if a space X of π-weight $ω_{1}$ is an absolute retract in the class of all extremally disconnected compact spaces and X is homogeneous with respect to π-weight (i.e. all non-empty open sets...

Nonseparable Radon measures and small compact spaces

Grzegorz Plebanek (1997)

Fundamenta Mathematicae

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We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube ${[0, 1]}^{κ}$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of $ω_{1}$ null sets in $2^{ω 1}$ such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is “no” for κ = ω. We also give alternative...

On partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1994)

Fundamenta Mathematicae

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We consider a set, L, of lines in $ℝ^{n}$ and a partition of L into some number of sets: $L = L_{1} \cup . . . \cup L_{p}$ . We seek a corresponding partition $ℝ^{n} = S_{1} \cup . . . \cup S_{p}$ such that each line l in $L_{i}$ meets the set $S_{i}$ in a set whose cardinality has some fixed bound, $ω_{τ}$ . We determine equivalences between the bounds on the size of the continuum, $2^{ω} \leq ω_{θ}$ , and some relationships between p, $ω_{τ}$ and $ω_{θ}$ .

Rigid $ℵ_{ε}$ -saturated models of superstable theories

Ziv Shami, Saharon Shelah (1999)

Fundamenta Mathematicae

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In a countable superstable NDOP theory, the existence of a rigid $ﬡ_{ε}$ -saturated model implies the existence of $2^{λ}$ rigid $ﬡ_{ε}$ -saturated models of power λ for every $λ > 2^{ﬡ_{0}}$ .

On absolutely divergent series

Sakaé Fuchino, Heike Mildenberger, Saharon Shelah, Peter Vojtáš (1999)

Fundamenta Mathematicae

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We show that in the $ℵ_{2}$ -stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under $c f () =$ the two algebras are isomorphic [15].

Bohr compactifications of discrete structures

Joan Hart, Kenneth Kunen (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and $1 \leq α < ω_{1}^{C K}$ , if for some $η < ℵ_{1}$ and all u ∈ WO of length η, a is $Σ_{α}^{0} (u)$ , then a is $Σ_{α}^{0}$ .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: $Σ_{1}^{1}$ -Turing-determinacy implies the existence of $0^{}$ .

On products of Radon measures

C. Gryllakis, S. Grekas (1999)

Fundamenta Mathematicae

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Let $X = {[0, 1]}^{Γ}$ with card Γ ≥ c (c denotes the continuum). We construct two Radon measures μ,ν on X such that there exist open subsets of X × X which are not measurable for the simple outer product measure. Moreover, these measures are strikingly similar to the Lebesgue product measure: for every finite F ⊆ Γ, the projections of μ and ν onto ${[0, 1]}^{F}$ are equivalent to the F-dimensional Lebesgue measure. We generalize this construction to any compact group of weight ≥ c, by replacing the Lebesgue product...

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

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We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\hat{X}$ is an ultrapower of X and B is a ball in $\hat{X}$ , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B. The preceding...

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

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Let X be a Polish space, and let ${(A_{p})}_{p \in ω}$ be a sequence of $G_{δ}$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether $\cup_{p \in ω} A_{p}$ is a true $\sum_{3}^{0}$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true $\sum_{3}^{0}$ .

Topological realization of a family of pseudoreflection groups

Dietrich Notbohm (1998)

Fundamenta Mathematicae

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We are interested in a topological realization of a family of pseudoreflection groups $G \subset G L (n, F_{p})$ ; i.e. we are looking for topological spaces whose mod-p cohomology is isomorphic to the ring of invariants $F_{p} {[x_{1}, . . ., x_{n}]}^{G}$ . Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over $F_{p}$ can appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product $ℤ / q ≀ Σ_{n}$ where q divides p - 1 and...

The theory of dual groups

A. Mekler, G. Schlitt (1994)

Fundamenta Mathematicae

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We study the $L_{\infty, w}$ -theory of sequences of dual groups and give a complete classification of the $L_{\infty, w}$ -elementary classes by finding simple invariants for them. We show that nonstandard models exist.

Countable partitions of the sets of points and lines

James Schmerl (1999)

Fundamenta Mathematicae

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The following theorem is proved, answering a question raised by Davies in 1963. If $L_{0} \cup L_{1} \cup L_{2} \cup . . .$ is a partition of the set of lines of $ℝ^{n}$ , then there is a partition $ℝ^{n} = S_{0} \cup S_{1} \cup S_{2} \cup . . .$ such that $| ℓ \cap S_{i} | \leq 2$ whenever $ℓ \in L_{i}$ . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

If it looks and smells like the reals...

Franklin Tall (2000)

Fundamenta Mathematicae

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Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_{M}$ is homeomorphic to ℝ, then $X = X_{M}$ . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

Definability within structures related to Pascal’s triangle modulo an integer

Alexis Bès, Ivan Korec (1998)

Fundamenta Mathematicae

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Let Sq denote the set of squares, and let $S Q_{n}$ be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let $B_{n} (x, y) = (\binom{x + y}{x}) M O D n$ . For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.

Hausdorff ’s theorem for posets that satisfy the finite antichain property

Uri Abraham, Robert Bonnet (1999)

Fundamenta Mathematicae

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Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC). Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations:...