Abstract

The minimum entropy degree one circle maps depending on the rotation interval are known since 1988 [ALMM]. In this very technical seminar I will obtain the intertwinning of the the two extremal twist orbits of these maps when the endpoints of the interval are rational and I will show that this information characterizes completely the dynamics of such maps. In a second part I will show how to extend these maps to an arbitrary graph with a single circuit by essentially keeping the dynamics. This is a factory of particular examples with (a limited) dynamics analogous to circle dynamics of the the minimum entropy degree one circle maps.

[ALMM] Lluı́s Alsedà, Jaume Llibre, Francesc Mañosas, and Michal Misiurewicz. Lower bounds of the topological entropy for continuous maps of the circle of degree one. Nonlinearity, 1(3):463–479, 1988.