Structures of R(f)−P(f)‾R(f)-\overline{P(f)} for graph maps ff

Let $G$ be a graph and $f: G\rightarrow G$ be a continuous map. We establish a structure theorem which describes the structures of the set $R(f)-\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set and the periodic point set of $f$ respectively. Roughly speaking, the set $R(f)-\overline{P(f)}$ is covered by finitely many pairwise disjoint $f$-invariant open sets $U_{1\,},\,\cdots,\,U_{n\,}$; each $U_i$ contains a unique minimal set $W_i$ which absorbs each point of $U_i$; each $W_i$ lies in finitely many pairwise disjoint circles each of which is contained in a connected closed set; all of these connected closed sets are contained in $U_i$ and permutated cyclically by $f$. As applications of the structure theorem, several known results are improved or reproved

SCRAMBLED SETS OF CONTINUOUS MAPS OF

Abstract. Let K be a 1-dimensional simplicial complex in R 3 without isolated vertexes, X = |K | be the polyhedron of K with the metric dK induced by K, andf:X→Xbe a continuous map. In this paper we prove that if K is finite, then the interior of every scrambled set of f in X is empty. We also show that if K is an infinite complex, then there exist continuous maps from X to itself having scrambled sets with nonempty interiors, and if X = R or R+, then there exist C ∞ maps of X with the whole space X being a scrambled set. 1

The nonexistence of expansive commutative group actions on Peano continua having free dendrites

AbstractA dendrite D in a metric space X is said to be free if there exists a connected open set U in X such that U¯=D. In this paper, we prove that there is no expansive commutative group action on any Peano continuum having a free dendrite. In particular, no 1-dimensional compact ANR admits an expansive commutative group action

Periodic orbits for multivalued maps with continuous margins of intervals

Let $I$ be a bounded connected subset of $\mathbb{R}$ containing more than one point, and ${\mathcal{L}}(I)$ be the family of all nonempty connected subsets of $I$. Each map from $I$ to ${\mathcal{L}}(I)$ is called a {multivalued map}. A multivalued map $F\colon I\rightarrow{\mathcal{L}}(I)$ is called a multivalued map with continuous margins if both the left endpoint and the right endpoint functions of $F$ are continuous. We show that the well-known Sharkovskiĭ theorem for interval maps also holds for every multivalued map with continuous margins $F\colon I\rightarrow{\mathcal{L}}(I)$, that is, if $F$ has an $n$-periodic orbit and $n\succ m$ (in the Sharkovskiĭ ordering), then $F$ also has an $m$-periodic orbit