We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.

We consider the following topological spaces: $\mathbf{O}=\{z\in ℂ:|z+i|=1\}$, ${\mathbf{O}}_3=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1],Imz\ge 0\}$, ${\mathbf{O}}_4=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1]\}$, ${\infty }_1=\mathbf{O}:|z-i|=1\}\cup \{z\in ℂ:z\in [0,1]\}$, ${\infty }_2={\infty }_1\cup \{z\in ℂ:z^2\in [0,1]\}$, et $\mathbf{T}=\{z\in ℂ:z=\cos (3\theta )e^{i\theta },0\le \theta \le 2\pi \}$. Set $E\in \{{\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2,\mathbf{T}\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per\left(f\right)$ the set of periods of all periodic points of $f$. The set $K\subset \mathbb{N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset Per\left(f\right)$, then $Per\left(f\right)=\mathbb{N}$. (2) If $S\subset \mathbb{N}$ is a set such that for every $E$ map $f$, $S\subset Per\left(f\right)$ implies $Per\left(f\right)=\mathbb{N}$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2$ and $\mathbf{T}$.

A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.

A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n\left(x\right),f^n\left(y\right))>c$. In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.

We consider dynamical systems of the form $(X,f)$ where $X$ is a compact metric space and $f:X\to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $\omega$-limit sets, answering a question by Will Brian.

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega$-limit set $\omega \left(Y\right)$ of $Y$ is the limit point of the sequence ${\{\left(\mathrm{C}lY\right)·[i,\infty )\}}_{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.

It is shown that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, it is shown that each Peano continuum having a free dendrite admits no expansive nilpotent group actions.

For the Abel equation on a real-analytic manifold a dynamical criterion of solvability in real-analytic functions is proved.