Cylinder cocycle extensions of minimal rotations on monothetic groups

Mieczysław K. Mentzen; Artur Siemaszko

Displaying similar documents to “Cylinder cocycle extensions of minimal rotations on monothetic groups”

Noninvertible minimal maps

Sergiĭ Kolyada, L&#039;ubomír Snoha, Sergeĭ Trofimchuk (2001)

Fundamenta Mathematicae

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For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{- 1} (A)$ share with A those topological properties which describe how large a set is. Using...

Uniqueness of minimal projections onto two-dimensional subspaces

Boris Shekhtman, Lesław Skrzypek (2005)

Studia Mathematica

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We prove that minimal projections from $L_{p}$ (1 < p < ∞) onto any two-dimensional subspace are unique. This result complements the theorems of W. Odyniec ([OL, Theorem I.1.3], [O3]). We also investigate the minimal number of norming points for such projections.

Induced subsystems associated to a Cantor minimal system

Heidi Dahl, Mats Molberg (2009)

Colloquium Mathematicae

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Let (X,T) be a Cantor minimal system and let (R,) be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system (Y,S) there exists a closed subset Z of X such that (Y,S) is conjugate to the subsystem (Z,T̃), where T̃ is the induced map on Z from T. We explore when we may choose Z to be a T-regular and/or a T-thin set, and we relate T-regularity of a set to R-étaleness. The latter concept plays an important role in the...

Forcing relation on minimal interval patterns

Jozef Bobok (2001)

Fundamenta Mathematicae

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Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by $h (g^{m} (m i n T)) = f^{m} (m i n S)$ is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing...

On groups of essential values of topological cylinder cocycles over minimal rotations

Mieczysław K. Mentzen (2003)

Colloquium Mathematicae

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A theory of essential values of cocycles over minimal rotations with values in locally compact Abelian groups, especially $ℝ^{m}$ , is developed. Criteria for such a cocycle to be conservative are given. The group of essential values of a cocycle is described.

Density of Morse functions on sets definable in o-minimal structures

Ta Lê Loi (2006)

Annales Polonici Mathematici

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We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable $C^{p}$ functions endowed with the Whitney topology.

Minimal actions of homeomorphism groups

Yonatan Gutman (2008)

Fundamenta Mathematicae

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Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ \subset 2^{2^{X}}$ , the space of maximal chains in $2^{X}$ , equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{H o m e o (X)}$ , the universal minimal space of Homeo(X), is not transitive...

Constants of strong uniqueness of minimal norm-one projections

A. Micek (2011)

Banach Center Publications

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In this paper we calculate the constants of strong uniqueness of minimal norm-one projections on subspaces of codimension k in the space $l_{\infty}^{(n)}$ . This generalizes a main result of W. Odyniec and M. P. Prophet [J. Approx. Theory 145 (2007), 111-121]. We applied in our proof Kolmogorov’s type theorem (see A. Wójcik [Approximation and Function Spaces (Gdańsk, 1979), PWN, Warszawa / North-Holland, Amsterdam, 1981, 854-866]) for strongly unique best approximation.

A characterization of a minimal submanifold in $R^{n + p}$

Themis Koufogiorgos (1983)

Annales Polonici Mathematici

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A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a ₁, . . ., a_{h}$ and negative terms $b ₁, . . ., b_{k}$ . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $σ ⁺ = \sum_{i = 1}^{h} a_{i} = - \sum_{j = 1}^{k} b_{j}$ . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive...

O-minimal version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (2014)

Studia Mathematica

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This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic $^{p}$ -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a $^{p}$ -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a...

Codimension one minimal projections in Banach spaces and a mathematical programming problem

Vladimir P. Odinec

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CONTENTSIntroduction.........................................................................................................5§1. Notation and preliminaries............................................................................6§2. Mathematical Programming Problem related to minimal projections.............8§3. Criteria for uniqueness of minimal projections in $l_{1}^{m}$ ..........................16References......................................................................................................27 ...

Weierstrass division theorem in definable $C^{\infty}$ germs in a polynomially bounded o-minimal structure

Abdelhafed Elkhadiri, Hassan Sfouli (2006)

Annales Polonici Mathematici

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We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty}$ functions.

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this article, we study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give...

A note on universal minimal dynamical systems

Sławomir Turek (1991)

Commentationes Mathematicae Universitatis Carolinae

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Let $M (G)$ denote the phase space of the universal minimal dynamical system for a group $G$ . Our aim is to show that $M (G)$ is homeomorphic to the absolute of $D^{2^{ω}}$ , whenever $G$ is a countable Abelian group.

Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio (2015)

Studia Mathematica

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We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection ${X_{i}}_{i \in I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $\prod_{i \in I} X_{i}$ is minimal if and only if $\prod_{i \in I} X_{i}^{e q}$ is minimal, where $X_{i}^{e q}$ is the maximal equicontinuous factor of $X_{i}$ . Most importantly, this result holds when each $X_{i}$ is distal. When...

The universal minimal system for the group of homeomorphisms of the Cantor set

E. Glasner, B. Weiss (2003)

Fundamenta Mathematicae

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Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E)...

Definable stratification satisfying the Whitney property with exponent 1

Beata Kocel-Cynk (2007)

Annales Polonici Mathematici

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We prove that for a finite collection of sets $A ₁, . . ., A_{s} \subset ℝ^{k + n}$ definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto $ℝ^{k}$ satisfy the Whitney property with exponent 1.

Minimal models for $ℤ^{d}$ -actions

Bartosz Frej, Agata Kwaśnicka (2008)

Colloquium Mathematicae

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We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$ -action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$ -action with the same, i.e. affinely homeomorphic, simplex of measures.

A uniform dichotomy for generic $SL (2, ℝ)$ cocycles over a minimal base

Artur Avila, Jairo Bochi (2007)

Bulletin de la Société Mathématique de France

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We consider continuous $SL (2, ℝ)$ -cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

Topological friction in aperiodic minimal $ℝ^{m}$ -actions

Jarosław Kwapisz (2010)

Fundamenta Mathematicae

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For a continuous map f preserving orbits of an aperiodic $ℝ^{m}$ -action on a compact space, its displacement function assigns to x the “time” $t \in ℝ^{m}$ it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by $_{V} (X, Z) : = {P \in (X, Z) : P |}_{V} = i d$ . Now let X = c₀ or $l_{\infty}^{m}$ , Z:= kerf for some f ∈ X* and $V : = Z \cap l ⁿ_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and...

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

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Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $γ \mapsto o r d (R e s (φ^{γ}))$ factors through a function $o r d R e s_{φ} (\cdot)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P ¹_{K}$ , or on a segment, and the minimal resultant locus is contained in the tree in $P ¹_{K}$ spanned by the fixed points...

Minimal systems and distributionally scrambled sets

Piotr Oprocha (2012)

Bulletin de la Société Mathématique de France

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In this paper we investigate numerous constructions of minimal systems from the point of view of $(ℱ_{1}, ℱ_{2})$ -chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$ ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$ -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results...

Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert Eyabi, Jobby Jacob, Renu C. Laskar, Darren A. Narayan, Dan Pillone (2012)

Discussiones Mathematicae Graph Theory

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For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, $ψ_{r} (G)$ , of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.

Expansions of o-minimal structures by sparse sets

Harvey Friedman, Chris Miller (2001)

Fundamenta Mathematicae

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Given an o-minimal expansion ℜ of the ordered additive group of real numbers and E ⊆ ℝ, we consider the extent to which basic metric and topological properties of subsets of ℝ definable in the expansion (ℜ,E) are inherited by the subsets of ℝ definable in certain expansions of (ℜ,E). In particular, suppose that $f (E^{m})$ has no interior for each m ∈ ℕ and $f : ℝ^{m} \to ℝ$ definable in ℜ, and that every subset of ℝ definable in (ℜ,E) has interior or is nowhere dense. Then every subset of ℝ definable in (ℜ,(S))...

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends

Gerd Dethloff, Pham Hoang Ha, Pham Duc Thoan (2016)

Colloquium Mathematicae

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We study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

On the Euler characteristic of the links of a set determined by smooth definable functions

Krzysztof Jan Nowak (2008)

Annales Polonici Mathematici

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The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ( $C^{\infty}$ ) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism....