@article{Alsedà_Bordignon_Groisman_2024, title={Topological entropy, sets of periods, and transitivity for circle maps}, volume={76}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7659}, DOI={10.3842/umzh.v76i1.7659}, abstractNote={<p>UDC 517.9</p> <p>Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems.<span class="Apple-converted-space"> </span>It is known that, for every graph that is not a tree and any $\varepsilon>0,$ there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than $\varepsilon$ (simplicity).<span class="Apple-converted-space"> </span>To numerically measure the complexity of the set of periods, we introduce a notion of the <em>boundary of cofiniteness.</em><span class="Apple-converted-space"> </span>Larger boundary of cofiniteness corresponds to a simpler set of periods.<span class="Apple-converted-space"> </span>We show that, for any continuous degree one circle maps, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness.</p>}, number={1}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Alsedà, Lluís and Bordignon, Liane and Groisman, Jorge}, year={2024}, month={Feb.}, pages={31 - 47} }