The research was conducted in collaboration with specialists from both the host institution (AGH University of Science and Technology, Kraków, PL) and abroad (Matej Bel University, Banska Bystrica, SK; Silesian University in Opava, CZ; University of Ostrava, CZ). During 18 months of the period Jana was working very hard for solutions of the problems set up in the project application. Together with Piotr Oprocha, they set up research plan, which resulted in many interesting solutions and generated additional open questions for research. The outcomes were summarized in 4 research articles published (or submitted for review) in peer-reviewed journals with well established position in the field of topology and dynamical systems:
In the article “On the structure of α-limit sets of backward trajectories for graph maps” by M. Foryś-Krawiec, J. Hantáková and P. Oprocha we study what sets can be obtained as α-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those α-limit sets are ω-limit sets and for all but finitely many points x, we can obtain every ω-limits set as the α-limit set of a backward trajectory starting in x. For zero entropy maps, every α-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization of α-limit sets of backward trajectories which is very close to complete picture of the possible situations.
The article “On backward attractors of interval maps” by J. Hantáková and S. Roth focusses on special α-limit sets (sα-limit sets). We explore the notion of sα-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz and Snoha. We give a criterion in terms of Xiong's attracting center that completely characterizes which interval maps have all sα-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. Since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors.
The article “Spaces where all closed sets are α-limit sets” by J. Hantáková, S. Roth and L. Snoha investigates the α-limit sets in more general topological spaces. Metrizable spaces are studied in which every closed set is an α-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs. Further it is shown that this property is not preserved by topological sums, products and continuous images and quotients. However, positive results do hold for metrizable spaces obtained by those constructions from spaces with enough arcs.
In the article “Dendrites and measures with discrete spectrum” by M. Foryś-Krawiec, J. Hantáková, J. Kupka, P. Oprocha and S. Roth we study dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrite is countable.
The results are of great interest for community working with topological dynamics. Jana presented the conducted work to specialized audience of experts in the field of discrete dynamical systems on the 9th Visegrad Conference: Dynamical Systems, Prague 2021. She also communicated the findings of this project with research groups at the host university on the Dynamical Systems Seminar on October 15, 2021, and on the Seminar of the Department of Differential Equations on April 27, 2021. She presented obtained results during three research visits abroad - Matej Bel University, Banska Bystrica, SK, in August 2020; Silesian University in Opava, CZ, in September 2021; University of Vienna, AT, in October 2021. Dissemination of mathematical findings to general public was obtained through participation in the European Researcher’s Night 2021 on September 24, 2021, with a talk “Henri Poincaré in the world of Game of Thrones: How long is the winter in Westeros”. The aim of this talk was presentation of intriguing world of topological dynamics, chaos theory (including own work of Jana) in accessible and informal fashion to public audience. Jana also organized a project day on fractal geometry and its applications on November 4, 2021, for students from high school Wichterlovo gymnázium, Ostrava, CZ.
