We prove that for continuous interval maps the existence of a non-empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.
For a continuous map on a topological graph containing a loop it is possible to define the degree (with respect to the loop ) and, for a map of degree , rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop then the set of rotation numbers of points in has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational...
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