Complexity and Simplicity in the dynamics of Totally Transitive graph maps [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Liane Bordignon
Universidade Federal de Sao Carlos


Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. In this talk we will discuss some relations between transitivity, periodic points, set of periods and topological entropy of continuous maps on graphs which are not trees. We will give examples of sequences of totally transitive graph maps $(f_n)$, $n \in \mathbb{N}$ with topological entropy going to zero when $n$ goes to $\infty$. We define \emph{boundary of cofiniteness} to measure complexity of the set of periods of a map and to speak about its density in $\mathbb{N}$. In the families in our examples, the boundary of cofiniteness goes to infinity while all the maps in the sequence have periodic points with small periods. This points out that the entropy may not be a good measure of dynamical complexity. We construct these examples in the circle and then show how to transfer them to any graph that are not a tree. Moreover, we show that in the circle, if the entropy of a sequence of transitive maps with periodic points goes to zero, then the boundary of cofiniteness goes to infinity and the diameter of the rotation interval goes to zero.

This is a joint ongoing work with Ll. Alsedà and J. Groisman.