Unique ergodicity for zero-entropy dynamical systems with the approximate product property. (English) Zbl 1466.37011
Summary: We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37B40 | Topological entropy |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37E05 | Dynamical systems involving maps of the interval |
| 37A25 | Ergodicity, mixing, rates of mixing |
Keywords:
approximate product property; unique ergodicity; topological entropy; ergodic measure; minimality; specification; gluing orbit; interval map; periodic pointsReferences:
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