Related papers: Equicontinuity of maps on dendrites

We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural…

Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. We…

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps…

We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of…

We prove that for every integer sequence $I$ satisfying Dold relations there exists a map $f : \mathbb{R}^d \to \mathbb{R}^d$, $d \ge 2$, such that $\mathrm{Per(f)} = \mathrm{Fix(f)} = \{o\}$, where $o$ denotes the origin, and $(i(f^n,…

We confirm the equitable $\Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or…

For metric spaces $X$ and $Y$, normed spaces $E$ and $F$, and certain subspaces $A(X,E)$ and $A(Y,F)$ of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps $T:A(X,E)\to A(Y,F)$ preserving…

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive…

In this paper we prove that for functions $f\in A(D)$ there is an equivalence between the continuous extension of their derivatives over the boundary and the differentiability of the map $t\mapsto f(e^{it})$. More specifically, we are able…

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a…

We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve the product ordering of R^n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the…

This paper gives an alternate, elementary proof of a result of Magnani: maps between Carnot groups that preserve horizontal curves and are continuously differential in horizontal directions in the Euclidean sense are continuously Pansu…

We prove that if $X$ is a paracompact connected space and $Z=\prod_{s\in S}Z_s$ is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map $g:X\to Z$ there exists a separately…

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…

We prove that, for every decreasing sequence {a \sb k} of natural numbers, there exists a map f: X --> X with cat (f\sp k)=a\sb k.

We give a short proof that for a bounded domain $\Omega\subset\mathbb{R}^n$ and continuous boundary data $g\in C(\partial\Omega)$ admitting a continuous finite-energy extension $\phi\in H^{1}(\Omega)\cap C(\bar\Omega)$, the minimizer of the…

The paper studies the structure of $\omega$-limit sets of map $\tilde{f}$ induced on the hyperspace $C(G)$ of all connected compact sets, by dynamical system $(G,f)$ acting on a topological graph $G$. In the case of the base space being a…

In this paper, we introduce the notion of negatively regionally proximal pairs of onto maps which coincides with the set of regionally proximal pair of $f^{-1}$, whenever $f$ is an homeomorphism and we prove the maximal equicontinoues…

Let $f:X\rightarrow Y$ be a map. $f$ is a {\it sequence-covering map}\cite{Si1} if whenever $\{y_{n}\}$ is a convergent sequence in $Y$ there is a convergent sequence $\{x_{n}\}$ in $X$ with each $x_{n}\in f^{-1}(y_{n})$; $f$ is an {\it…

Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C.…