Topology and its Applications最新文献

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset $G\subset C$. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set G of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set $M_n$ for fractal n-gons. We prove that $M_n$ is connected and locally connected for any n.

In this paper, we study the notions of $F$-sensitivity, multi-$F$-sensitivity and $(F_1,F_2)$-sensitivity in dynamical systems defined on Hausdorff uniform spaces. It is shown that how these notions carry over to hyperspatial and product dynamical systems, and vice versa. We also obtain some sufficient conditions for a dynamical system to be $F$-sensitive. Some examples showing the necessity of the conditions taken in our results are also presented.

Let $C\left(X\right)$ be the hyperspace of all subcontinua of a continuum X topologized by Hausdorff metric. For a non-empty closed subset A of a continuum X, consider the following properties: A is a strong non-cut subset, non-block subset, weak non-block subset, shore subset, not a strong center, and non-cut subset of X. The aim of this paper is to study the conditions under which an ordered arc from a singleton to a proper subcontinuum of a continuum X has one of these properties in $C\left(X\right)$.

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups $f:(\Xi ,{\Xi }^{\infty })\to (\Lambda ,{\Lambda }^{\infty })$, stating that $\mathrm{asdim}\Xi \le \mathrm{asdim}\Lambda +\mathrm{asdim}\left({[\ker f]}_c\right)$. This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group G, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if $(\Lambda ,{\Lambda }^{\infty })$ is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of ${\Lambda }^{\infty }$, with these approximate subgroups contained in ${\Lambda }^2$.

Let $P_k$ be the graded polynomial algebra $F_2[x_1,x_2,\dots ,x_k]$ over the prime field of two elements, $F_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for $P_k$ as a module over the mod-2 Steenrod algebra, $A$. In this paper, we explicitly determine a minimal set of $A$-generators for $P_5$ in the case of the generic degrees $n=2^{d+1}-1$ and $n=2^{d+1}-2$ for all $d⩾6$.

Let $(X,d)$ be a metric space and $f_{1,\infty }={\left\{f_n\right\}}_{i=0}^{\infty }$ be a sequence of continuous maps $f_n:X\to X$ such that $\left(f_n\right)$ converges uniformly to a continuous map f. We investigate which conditions ensure that the transitivity of functions $f_n$ or the transitivity of the nonautonomous system $(X,f_{1,\infty })$ is inherited to the limit function f and vice versa. Such problem has been studied for instance by A. Fedeli, A. Le Donne or J. Li who give different sufficient condition for inheriting of transitivity from $f_n$ to f. In this paper we give a survey of known result relating to this problem and prove new results concerning transitivity.

We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed graph theorem, and we compare closed graph and open mapping spaces.

It is well known that the Hausdorff dimension and box dimension of a Bedford-McMullen carpet coincide if and only if the carpet has uniform horizontal fibers. Yang et al. [17] gave a complete Lipschitz classification of such carpets which are totally disconnected and possessing vacant rows. In this paper, we show that if we replace the uniform horizontal fibers condition by a locally uniform horizontal fibers condition, the conclusions of Yang et al. still hold.

We present a property equivalent to the property of being semi-Kelley. Using this equivalence we prove that being semi-Kelley is a hereditary property for atriodic continua. We prove that semi-Kelley remainders are atriodic, moreover, we prove that semi-Kelley continua are semi-Kelley remainders for chainable continua, circularly chainable continua, and arc continua, and we give an example of an atriodic Kelley continuum which is a semi-Kelley remainder and not a Kelley remainder. We also prove that hereditarily semi-Kelley dendroids are smooth.