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

Date:
02.05.16   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Lluis Alsedà
University:
Universitat Autònoma de Barcelona

Abstract:

Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon > 0$, there exist (complicate) totally transitive map (then with cofinite set of periods) such that the topological entropy is smaller than $\varepsilon$ (simplicity). In the first part of the talk, delivered by Liane Bordignon, it was shown by means of examples and for the circle that the above scenario: for graphs that are not trees there exist relatively simple maps (with small entropy) which are totally transitive (and hence robustly complicate) can be extended to the set of periods. To measure numerically the complexity of the set of periods we introduce the notion of boundary of cofiniteness defined as the smallest positive integer $n$ such that the set of periods contains $\{n, n+1, n+2, \dots\}$. Larger boundary of cofiniteness means simpler set of periods. With the help of the notion of boundary of cofiniteness we can state precisely what do we mean by extending the entropy simplicity result to the set of periods: \emph{there exist relatively simple maps such that the boundary of cofiniteness is arbitrarily large (simplicity) which are totally transitive (and hence robustly complicate)}. In the first part of the talk several examples in arbitrary graphs were discussed and it was shown that for circle maps the above statement is a theorem. In this talk we will extend this statement to the space $\sigma$ and we will discuss its proof. This is a good example on how the lack of knowledge about the structure of the set of periods can be overcome with appropriate simple arguments.

This is a joint ongoing work with L. Bordignon and J. Groisman.