The nonexistence of expansive \(\mathbb Z^n\) actions on Peano continua that contain no \(\theta\)-curves
This paper deals with topological dynamics aspects of homeomorphisms defined on Peano continua. Let the reviewer introduce some definitions in order to state the main results of this paper. By \(X\) we denote a compact metric space, \(G\) is a topological group with discrete topology, and \(\text{Hom}(X)\) is the group of homeomorphisms from \(X\) into itself. An action of \(G\) on \(X\) is a group homeomorphism \(\pi :G\rightarrow \text{Hom}(X)\). If \(G=\mathbb{Z}\) and \(\pi (n)=f^{n}\) where \(f\) is a fixed homeomorphism on \(X\), we obtain the classical scheme of a discrete dynamical system. An action \(\pi \) is called expansive if there exists a positive constant \(c\) such that for any pair of different points \(x,y\) from \(X,\) there exists a \(g\in G\) such that \(d(\pi (g)(x),\pi (g)(y))>c.\) The authors study the (non) existence of expansive actions on Peano continua (locally connected, compact, and connected metric spaces). The main results are the following. Theorem 2.11. There are no expansive \(\mathbb{Z}^{n}\) actions on dendrites (Peano continua containing no simple closed curves). Theorem 2.12. Peano continua that contain no \(\theta \)-curves do not admit expansive \(\mathbb{Z}^{n}\) actions. (Roughly speaking, a \(\theta \)-curve is a compact connected space having the form of letter \(\theta \)). It is interesting to mention that the authors make use of a result of \textit{J. R. Isbell} [Res. Probl. 7, Bull. Am. Math. Soc. 63, 419 (1957)], which states that every commutative group \(\Gamma \) acting on a dendrite \(X\) has a fixed point, that is, there exists \(x\in X\) such that \(\pi (g)(x)=x\) for all \(g\in \Gamma. \) With this property they are able to find a subgroup \(H\) of \(G=\mathbb{Z}^{n}\) such that \(\pi (H)(A)=A\) where \(A\subset X\) is a closed arc or an infinite-type star-like dendrite. Finally, they propose the following question: does there exist an expansive \(\mathbb{Z}^{n}\) action on any Peano continuum contained in the plane?
scientific article; zbMATH DE number 5123631
scientific article