Displaying similar documents to “On the diophantine equation n k = x l

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

Sebastian van Strien (2000)

Fundamenta Mathematicae

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We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if f λ 0 is critically finite with non-degenerate critical point c 1 ( λ 0 ) , . . . , c n ( λ 0 ) such that f λ 0 k i ( c i ( λ 0 ) ) = p i ( λ 0 ) are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31   λ ( f λ k 1 ( c 1 ( λ ) ) - p 1 ( λ ) , . . . , f λ k d - 2 ( c d - 2 ( λ ) ) - p d - 2 ( λ ) ) is a local diffeomorphism...

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...

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 α < ω 1 C K , if for some η < 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 .

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 and a sequence of continuous maps ( f i ) i = 1 , f i : X i 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 and...

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 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 " α ω ω , ω ω L ( α ) is bounded for ≤*".

The theory of dual groups

A. Mekler, G. Schlitt (1994)

Fundamenta Mathematicae

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

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...

On the equation a p + 2 α b p + c p = 0

Kenneth A. Ribet (1997)

Acta Arithmetica

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We discuss the equation a p + 2 α b p + c p = 0 in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.

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...

The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo, Salvador Hernández (1999)

Fundamenta Mathematicae

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Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification....

On absolutely divergent series

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

Fundamenta Mathematicae

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We show that in the 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].

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 ...

Properties of the class of measure separable compact spaces

Mirna Džamonja, Kenneth Kunen (1995)

Fundamenta Mathematicae

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We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular)...

The sequential topology on complete Boolean algebras

Wiesław Główczyński, Bohuslav Balcar, Thomas Jech (1998)

Fundamenta Mathematicae

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We investigate the sequential topology τ s on a complete Boolean algebra B determined by algebraically convergent sequences in B. We show the role of weak distributivity of B in separation axioms for the sequential topology. The main result is that a necessary and sufficient condition for B to carry a strictly positive Maharam submeasure is that B is ccc and that the space ( B , τ s ) is Hausdorff. We also characterize sequential cardinals.

Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Feliks Przytycki (1994)

Fundamenta Mathematicae

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We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. f - 1 ( A ) = A ), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neigh bourhoods arrive at large scale” under iteration of f. This generalizes...

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...