Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio

Displaying similar documents to “Multiple disjointness and invariant measures on minimal distal flows”

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.

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 $ℝ^{4}$ .

An attraction result and an index theorem for continuous flows on $ℝ^{n} \times [0, \infty)$

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

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We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n + 1}$ for which $\partial E = ℝ^{n} \times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^{n} \times [0, \infty)$ .

Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker (2014)

Fundamenta Mathematicae

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In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $G L (V_{\infty})$ , where $V_{\infty}$ is the countably...

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...

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

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We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

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...

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_{G}$ . We consider the map $τ_{φ}$ defined on X × G by $τ_{φ} : (x, g) \mapsto (τ x, φ (x) g)$ and the cocycle ${(φ ₙ)}_{n \in ℤ}$ generated by φ. Using a characterization of the ergodic invariant measures for $τ_{φ}$ , we give the form of the ergodic decomposition of $μ (d x) \otimes m_{G} (d g)$ or more generally of the $τ_{φ}$ -invariant measures $μ_{χ} (d x) \otimes χ (g) m_{G} (d g)$ , where $μ_{χ} (d x)$ is χ∘φ-conformal for an exponential χ on G.

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...

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.

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

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...

Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang (2008)

Studia Mathematica

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We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$ , there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$ . Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang (2021)

Czechoslovak Mathematical Journal

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We construct some nondecreasing quantities associated to the first eigenvalue of $- Δ_{φ} + c R$ $(c \geq \frac{1}{2} (n - 2) / (n - 1))$ along the Yamabe flow, where $Δ_{φ}$ is the Witten-Laplacian operator with a $C^{2}$ function $φ$ . We also prove a monotonic result on the first eigenvalue of $- Δ_{φ} + \frac{1}{4} (n / (n - 1)) R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $- Δ_{φ} + c R^{a}$ with $a \in (0, 1)$ along the Yamabe flow.

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

Geometric rigidity of $\times m$ invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let $μ$ be a probability measure on $[0, 1]$ which is invariant and ergodic for $T_{a} (x) = a x 𝚖𝚘𝚍 1$ , and $0 < 𝚍𝚒𝚖 μ < 1$ . Let $f$ be a local diffeomorphism on some open set. We show that if $E \subseteq ℝ$ and $(f μ) {|_{E} \sim μ|}_{E}$ , then $f^{'} (x) \in \{\pm a^{r} : r \in ℚ\}$ at $μ$ -a.e. point $x \in f^{- 1} E$ . In particular, if $g$ is a piecewise-analytic map preserving $μ$ then there is an open $g$ -invariant set $U$ containing supp $μ$ such that $g |_{U}$ is piecewise-linear with slopes which are rational powers of $a$ . In a similar vein, for $μ$ as above, if $b$ is another integer and $a, b$ are not powers of a common integer, and if $ν$ is...

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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Invariant theory and the $𝒲_{1 + \infty}$ algebra with negative integral central charge

Andrew Linshaw (2011)

Journal of the European Mathematical Society

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The vertex algebra $𝒲_{1 + \infty, c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n \geq 1$ , it was conjectured in the physics literature that $𝒲_{1 + \infty, - n}$ should have a minimal strong generating set consisting of $n^{2} + 2 n$ elements. Using a free field realization of $𝒲_{1 + \infty, - n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation...

Fraïssé structures and a conjecture of Furstenberg

Dana Bartošová, Andy Zucker (2019)

Commentationes Mathematicae Universitatis Carolinae

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We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S (G)$ , the Samuel compactification, and $E (M (G))$ , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of $G = S_{\infty}$ , leading us to define and investigate several...

Three examples of brownian flows on $ℝ$

Yves Le Jan, Olivier Raimond (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $d X_{t} = 1_{{X_{t} g t; 0}} W^{+} (d t) + 1_{{X_{t} l t; 0}} d W^{-} (d t),$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by $ϕ^{\pm}$ . The flow $ϕ^{\pm}$ is a Wiener solution of the SDE. Moreover, $K^{+} = 𝖤 [δ_{ϕ^{\pm}} | W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $K_{s, t}^{+} f (x) = f (x) + \int_{s}^{t} K_{s, u} (1_{ℝ}^{+} f^{'}) (x) W^{+} (d u) + \frac{1}{2} \int_{s}^{t} K_{s, u} f ` ` (x) d u$ for $s l t; t$ , $x \in ℝ$ and $f$ a twice continuously differentiable function. A third flow $ϕ^{+}$ can be constructed out of the $n$ -point motions of $K^{+}$ . This flow is coalescing and its $n$ -point motion...

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

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Let $U ₁, . . ., U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ ₊^{d}$ and the associated collection of rectangular parallelepipeds in $ℝ^{d}$ with sides parallel to the axes and dimensions of the form $n ₁ \times \dots \times n_{d}$ with $(n ₁, . . ., n_{d}) \in Γ .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{Γ}$ are defined respectively on $L ¹ (ℝ^{d})$ and L¹(Ω) by $M_{} g (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | g (y) | d y$ and $M_{Γ} f (ω) = s u p_{(n ₁, . . ., n_{d}) \in Γ} 1 / n ₁ \dots n_{d} \sum_{j ₁ = 0}^{n ₁ - 1} \dots \sum_{j_{d} = 0}^{n_{d} - 1} | f (U ₁^{j ₁} \dots U_{d}^{j_{d}} ω) |$ . Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${ϕ |}_{X ₀}$ is continuous in the relative topology on X₀ and $ϕ^{- 1} |_{Y ₀}$ is continuous in the relative topology on Y₀, (2) $ϕ (O r b_{T} (x)) = O r b_{S} (ϕ (x))$ for μ-a.e. x ∈ X. (X,,μ,T)...

O-minimal fields with standard part map

Jana Maříková (2010)

Fundamenta Mathematicae

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Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let $k_{i n d}$ be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in $k_{i n d}$ and conditions on (R,V) which imply o-minimality of $k_{i n d}$ . We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in $k_{i n d}$ are exactly the standard parts of the sets definable in (R,V).

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

Invariant densities for random $β$ -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ (β - 1)]$ . We show existence and uniqueness of a $K_{β}$ -invariant probability measure, absolutely continuous with respect to $m_{p} \otimes λ$ , where $m_{p}$ is the Bernoulli measure on ${0, 1}^{ℕ}$ with parameter $p$ ( $0 < p < 1$ ) and $λ$ is the normalized Lebesgue measure on $[0, ⌊ β ⌋ (β - 1)]$ . Furthermore, this measure is of the form $m_{p} \otimes μ_{β, p}$ , where $μ_{β, p}$ is equivalent to $λ$ . We prove that the measure of maximal entropy and $m_{p} \otimes λ$ are mutually...

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.

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

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A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space $_{φ}$ has pure discrete spectrum, we describe the collection $_{φ}^{P}$ of pairs of proximal tilings in $_{φ}$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then $_{φ}$ and $_{ψ}$ are homeomorphic...

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...

The group ring $𝕂 F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals

John Donnelly (2019)

Archivum Mathematicum

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We use a total order on Thompson’s group $F$ to show that the group ring $𝕂 F$ has no minimal non-zero ideals.

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^{p} (ν)$ , $T ₁, . . ., T_{k}$ , $n ̅ = (n ₁, . . ., n_{k}) \in ℕ^{k}$ and $α ̅ = (α ₁, . . ., α_{k})$ with $0 < α_{j} \leq 1$ , we define the ergodic Cesàro-α̅ averages $_{n ̅, α ̅} f = 1 / (\prod_{j = 1}^{k} A_{n_{j}}^{α_{j}}) \sum_{i_{k} = 0}^{n_{k}} \dots \sum_{i ₁ = 0}^{n ₁} \prod_{j = 1}^{k} A_{n_{j} - i_{j}}^{α_{j} - 1} T_{k}^{i_{k}} \dots T ₁^{i ₁} f$ . For these averages we prove the almost everywhere convergence on X and the convergence in the $L^{p} (ν)$ norm, when $n ₁, . . ., n_{k} \to \infty$ independently, for all $f \in L^{p} (d ν)$ with p > 1/α⁎ where $α ⁎ = m i n_{1 \leq j \leq k} α_{j}$ . In the limit case p = 1/α⁎, we prove that the averages $_{n ̅, α ̅} f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ (1 / α ⁎, φ_{m - 1})$ with $φ ₘ (t) = t {(1 + l o g ⁺ t)}^{m}$ . To obtain the result in the limit case we need...

A note on functional tightness and minitightness of space of the $G$ -permutation degree

Dimitrios N. Georgiou, Nodirbek K. Mamadaliev, Rustam M. Zhuraev (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the functor of the permutation degree. Analytically: a) We introduce the notion of $τ$ -open sets and investigate some basic properties of them. b) We prove that if the map $f : X \to Y$ is $τ$ -continuous, then the map $S P^{n} f : S P^{n} X \to S P^{n} Y$ is also $τ$ -continuous. c) We show that the functor $S P^{n}$ preserves the functional tightness and the minimal tightness of compacts. d) Finally, we give some facts and properties...