Minimality, transitivity, mixing and topological entropy on spaces with a free interval (Q2874005)

In this paper, the authors systematically study dynamics of continuous maps on compact metrizable spaces containing a free or a disconnecting interval and compare the results with those arising from works on graphs or trees.NEWLINENEWLINEThere are three main results in this paper. The first two theorems deal with minimal systems and minimal sets, the third one is a dichotomy for transitive maps. Let \(X\) be a compact metrizable space with a free interval \(J\) and let \(f : X \longrightarrow X\) be a continuous map. The results can be stated as follows:NEWLINENEWLINENEWLINE-- If \(f\) is minimal, then \(X\) is a disjoint union of finitely many circles, \(X=\oplus_{i=0}^{n-1}\mathbb{S}_i^1\), which are cyclically permuted by \(f\) and, on each of them, \(f^n\) is topologically conjugate to the same irrational rotation.NEWLINENEWLINE NEWLINE-- Assume that \(M\) is a minimal set for \(f\) which intersects \(J\). Then exactly one of the following three statements holds. (1) \(M\) is finite; (2) \(M\) is a nowhere dense cantoroid; (3) \(M\) is a disjoint union of finitely many circles.NEWLINENEWLINENEWLINE-- Assume \(f\) is transitive, then exactly one of the following two statements holds. (1) The map \(f\) is relatively strongly mixing, non-invertible, has positive topological entropy and dense periodic points. (2) The space \(X\) is a disjoint union of finitely many circles, \(X=\oplus_{i=0}^{n-1}\mathbb{S}_i^1\), which are cyclically permuted by \(f\) and, on each of them, \(f^n\) is topologically conjugate to the same irrational rotation.NEWLINENEWLINEThe obtained results are potentially applicable in the study of dynamics on some classes of one-dimensional continua and some of them are even surprisingly nice.