We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...

By a dynamical system $(X,T)$ we mean the action of the semigroup $({\mathbb{Z}}^{+},+)$ on a metrizable topological space $X$ induced by a continuous selfmap $T:X\to X$. Let $M\left(X\right)$ denote the set of all compatible metrics on the space $X$. Our main objective is to show that a selfmap $T$ of a compact space $X$ is a Banach contraction relative to some $d_1\in M\left(X\right)$ if and only if there exists some $d_2\in M\left(X\right)$ which, regarded as a $1$-cocycle of the system $(X,T)\times (X,T)$, is a coboundary.

In this paper piecewise monotonic maps $T[0,1]\to [0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R\left(Q\right)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $\left(R\right(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary...

If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space ${}_{\Phi }$ factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.

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.