South African - Japanese Discrete Homotopy Meeting 2025 (Cape Town, South Africa, 10-11 September 2025)

10th of September 2025  - 10.40/11.20  

Venue: MAM111 - Department of Mathematics and Applied Mathematics - University of Cape Town

Speaker:   Marco Farotti (University of Camerino, Camerino, Italy)

Title:          Linear suborders in the chain components poset

Abstract: Topological dynamical relations, and in particular chain recurrence, are typically studied in continuous dynamical systems, or at least in systems with some form of weakened continuity. The general case in which no regularity assumption is made is less investigated. We study the chain components generated by a dynamical system (X,f) from the perspective of poset theory, considering both the case where f is a continuous map and the general case where no regularity assumptions are made. First, we show that the poset of chain components cannot contain a linearly ordered, densely ordered subset if the map is continuous. This result holds even without assuming compactness, provided that certain topological conditions are satisfied. Then we prove that every countable well-order admitting a maximum is the order-type of the poset of chain components of a continuous map on an interval. Finally, we prove that the property that every linearly ordered segment has a minimum is true also without any regularity assumption.