Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada; Michał Misiurewicz; L’ubomír Snoha

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Displaying similar documents to “Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval”

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

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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)

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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...

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

Sebastian van Strien (2000)

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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 $λ \mapsto (f_{λ}^{k_{1}} (c_{1} (λ)) - p_{1} (λ), . . ., f_{λ}^{k_{d - 2}} (c_{d - 2} (λ)) - p_{d - 2} (λ))$ is a local diffeomorphism...

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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 $ω_{θ}$ .

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 $1 \leq α < ω_{1}^{C K}$ , 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 absolutely divergent series

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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].

Hausdorff dimension and measures on Julia sets of some meromorphic maps

Krzysztof Barański (1995)

Fundamenta Mathematicae

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We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f (z) = h (e x p \frac{2 π i}{T} z)$ where h is a rational function or, equivalently, the maps $˜ f (z) = e x p (\frac{2 π i}{h} (z))$ . When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller...

All solenoids of piecewise smooth maps are period doubling

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Fundamenta Mathematicae

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We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_{1} < . . . < p_{n}$ is a periodic orbit of a continuous map f then there is a union set $q_{1}, . . ., q_{n - 1}$ of some periodic orbits of f such that $p_{i} < q_{i} < p_{i + 1}$ for any i.

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 theory of dual groups

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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.

Interpreting reflexive theories in finitely many axioms

V. Shavrukov (1997)

Fundamenta Mathematicae

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For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation ’F interprets R’ in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of $\prod_{1}$ (as well as $\sum_{1}$ ) sentences π such that GB interprets ZF+π is $Σ_{3}^{0}$ -complete.

Raising dimension under all projections

John Cobb (1994)

Fundamenta Mathematicae

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As a special case of the general question - “What information can be obtained about the dimension of a subset of $ℝ^{n}$ by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in $ℝ^{3}$ each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in $ℝ^{n}$ whose images contain open sets, expanding on a result of Borsuk.

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 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 $ℒ_{\infty}$ -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification....

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:...

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 $p r o - H_{0}$ ...

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...