arXiv subjects
Publications and source records attributed to Taras Banakh.
In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.
In this paper we detect topological Clifford semigroups which are embeddable into Tychonoff products of topological semilattices and cones over topological groups. Also we detect topological Clifford semigroups which embed into compact topological Clifford semigroups.
In 1995 in Kourovka notebook the second author asked the following problem: it is true that for each partition $G=A_1\cup\dots\cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups.
We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)<n$ for some cell $A_i$ of the partition. Here $cov(A)=\min\{|F|:F\subset G,\;G=FA\}$ is the covering number of $A$ in $G$. A similar result is proved also of partitions of $G$-spaces. This gives two partial answers to a problem of Protasov posed in 1995.
Given a finite subset $Σ\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(Σ;q)=\big\{\sum_{n=0}^\infty a_nq^n:(a_n)_{n\inω}\inΣ^ω\big\}$, which is the unique compact solution of the equation $K=Σ+qK$. The obtained results are applied to studying partial sumsets $E(x)=\big\{\sum_{n=0}^\infty x_n\varepsilon_n:(\varepsilon_n)_{n\inω}\in\{0,1\}^ω\big\}$ of some (multigeometric) sequences $x=(x_n)_{n\inω}$.
For every functional functor $F:Comp\to Comp$ in the category $Comp$ of compact Hausdorff spaces we define the notions of $F$-Dugundji and $F$-Milutin spaces, generalizing the classical notions of a Dugundji and Milutin spaces. We prove that the class of $F$-Dugundji spaces coincides with the class of absolute $F$-valued retracts. Next, we show that for a monomorphic continuous functor $F:Comp\to Comp$ admitting tensor products each Dugundji compact is an absolute $F$-valued retract if and only if the doubleton $\{0,1\}$ is an absolute $F$-valued retract if and only if some points $a\in F(\{0\})\subset F(\{0,1\})$ and $b\in F(\{1\})\subset F(\{0,1\})$ can be linked by a continuous path in $F(\{0,1\})$. We prove that for the functor $Lip_k$ of $k$-Lipschitz functionals with $k<2$, each absolute $Lip_k$-valued retract is openly generated. On the other hand the one-point compactification of any uncountable discrete space is not openly generated but is an absolute $Lip_3$-valued retract. More generally, each hereditarily paracompact scattered compact space $X$ of finite scattered height $n$ is an absolute $Lip_k$-valued retract for $k=2^{n+2}-1$.
Given a set $X$ and a family $G$ of self-maps of $X$, we study the problem of the existence of a non-discrete Hausdorff topology on $X$ with respect to which all functions $f\in G$ are continuous. A topology on $X$ with this property is called a $G$-topology. The answer is given in terms of the Zariski $G$-topology $ζ_G$ on $X$, that is, the topology generated by the subbase consisting of the sets $\{x\in X:f(x)\ne g(x)\}$ and $\{x\in X:f(x)\ne c\}$, where $f,g\in G$ and $c\in X$. We prove that, for a countable monoid $G\subset X^X$, $X$ admits a non-discrete Hausdorff $G$-topology if and only if the Zariski $G$-topology $ζ_G$ is non-discrete; moreover, in this case, $X$ admits $2^{\mathfrak c}$ hereditarily normal $G$-topologies.
This text contains lecture notes of the course taught to Ph.D. students of Jagiellonian University in Krakow on 25-28 November, 2013.
A family $\mathcal N$ of closed subsets of a topological space $X$ is called a {\em closed $k$-network} if for each open set $U\subset X$ and a compact subset $K\subset U$ there is a finite subfamily $\mathcal F\subset\mathcal N$ with $K\subset\bigcup\F\subset \mathcal N$. A compact space $X$ is called {\em supercompact} if it admits a closed $k$-network $\mathcal N$ which is {\em binary} in the sense that each linked subfamily $\mathcal L\subset\mathcal N$ is centered. A closed $k$-network $\mathcal N$ in a topological group $G$ is {\em invariant} if $xAy\in\mathcal N$ for each $A\in\mathcal N$ and $x,y\in G$. According to a result of Kubiś and Turek, each compact (abelian) topological group admits an (invariant) binary closed $k$-network. In this paper we prove that the compact topological groups $S^3$ and $\SO(3)$ admit no invariant binary closed $k$-network.
We prove that a separable Hausdorff topological space $X$ containing a cocountable subset homeomorphic to $[0,ω_1]$ admits no separately continuous mean operation and no diagonally continuous $n$-mean for $n\ge 2$.
We introduce the Solecki submeasure $σ(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological groups. Also we show that the right Solecki density on a countable amenable group coincides with the upper Banach density $d^*$ which allows us to generalize some fundamental results of Bogoliuboff, Folner, Cotlar and Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson and Fish about the sumsets to the class of all amenable groups.
We prove that for every number k each countable infinite group $G$ admits a partition $G=A\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $μ:\mathcal P(G)\to[0,1]$ such that for each $ε> 1/|G|$ and subset $A\subset G$ with $μ(A)<1$ there is a set $B\subset G\setminus A$ such that $μ(B)<ε$ and $FB=G$ for some finite subset $F\subset G$.
In this paper we introduce and study a new topologo-algebraic structure called a (di)topological unosemigroup. This is a topological semigroup endowed with continuous unary operations of left and right units (which have certain continuous division property called the dicontinuity). We show that the class of ditopological unosemigroups contains all topological groups, all topological semilattices, all uniformizable topological unoid-semigroups, all compact topological unosemigroups, and is closed under the operations of taking subunosemigroup, Tychonoff product, reduced product, semidirect product, and the Hartman-Mycielski extension.
We prove that every infinite-dimensional (locally convex) linear topological space that can be expressed as a direct limit of finite-dimensional metrizable compacta is (linearly) homeomorphic to the space $R^\infty=\dlim R^n$.
Given a multi-valued function $Φ$ on a topological space $X$ we study the properties of its fixed fractal, which is defined as the closure of the orbit $Φ^ω(Fix(Φ))=\bigcup_{n\inω}Φ^n(Fix(Φ))$ of the set $Fix(Φ)=\{x\in X:x\inΦ(x)\}$ of fixed points of $Φ$. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals for a contracting compact-valued function $Φ$ on a complete metric space $X$ and its inverse multi-function $Φ^{-1}$. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpinski triangle, Sierpinski carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.
We prove that for each meager relation $E\subset X\times X$ on a Polish space $X$ there is a nowhere meager subspace $F\subset X$ which is $E$-free in the sense that $(x,y)\notin E$ for any distinct points $x,y\in F$.
Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into $n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set $F\subset G$ of cardinality $|F|\le ϕ(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot Δ(A_i)$ where $Δ(A_i)=\{g\in G:gA_i\cap A_i\notin I\}$ is the difference set of the set $A_i$. Also we investigate the growth of the sequence $ϕ(n)=\max_{1<x<n}\frac{x^{n-x}-1}{x-1}$ and show that $\ln ϕ(n)=nW(ne)-2n+\frac{n}{W(ne)}+\frac{W(ne)}{n}+O\big(\frac{\ln n}n\big)$ where $W(x)$ is the Lambert W-function, defined implicitly as $W(x)e^{W(x)}=x$. This shows that $ϕ(n)$ grows faster that any exponent $a^n$ but slower than the sequence of factorials $n!$.
We characterize semigroups $X$ whose semigroups of filters $φ(X)$, maximal linked systems $λ(X)$, linked upfamilies $N_2(X)$, and upfamilies $\upsilon(X)$ are commutative.