## Periodic Reporting for period 1 - LISEDIDYS (Limit sets of discrete dynamical systems)

Reporting period: 2020-06-01 to 2021-11-30

In real life we often encounter problems whose evolution can be defined in fixed state space according to a fixed rule. Populations, epidemics and economies at all scales are such examples, which mathematicians are trying to model through so-called dynamical systems. In the project we are interested in discrete dynamical systems, that is systems which evolve in discrete time steps. The aim of the project is studying long time behaviour of such systems (expressed in terms of the limit sets) in the future or the past. As one of research goals, we investigate structure, topological size and other topological and dynamical properties of limit sets. We focus on low dimensional dynamical systems such as interval maps, maps acting on topological graphs, dendrites, etc. While the limit sets of forward trajectories were deeply studied in past decades, the limit sets of backward trajectories, which should be regarded as sources of trajectories, have not been that much explored so far. Our aim was obtaining good characterization of the properties of limit sets of backward trajectories for interval and more widely, graph maps or even maps acting on general topological spaces. In the context of forward dynamics, good characterizations exist on topological graphs, however dynamical properties of maps on dendrites are not completely characterized. Therefore, another goal was understanding of limit sets of forward trajectories for maps acting on dendrites and related open questions from measure theory about spectrum of invariant measures when topological entropy is zero.

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.

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.

The goal of the project was to explore different concepts of limit sets in order to improve our understanding of complex dynamics and nonlinear phenomena. At the beginning of the project, not much was known about limit sets of backward trajectories in low dimensional topological dynamics. We obtained almost complete picture of limit sets of backward trajectories for maps on topological graphs. We also shaped a general theory on various types of limit sets of backward trajectories in much more universal spaces. Finally, our solution on spectrum of measures of zero-entropy maps on dendrites solves a problem which was open for a while.

When it comes to a wider impact of our results, they will be exploited primarily for further progress by the research groups and the wider topological dynamics community. A profound study of limit sets will contribute to our knowledge of long-term behaviour of discrete dynamical systems. It is known that rendering long-term prediction of their behaviour by computer is impossible in general, even though these systems are deterministic. Limit sets can be very complicated, this is often the case when the dynamics is chaotic. Chaotic behaviour exists in many natural dynamical systems, continuous or discrete, such as weather and climate or biological model of population growth, and chaos theory has applications in several disciplines, including meteorology, physics, computer science, economics and biology. These applications are a long-term exploitation of the fundamental research of dynamical systems.

When it comes to a wider impact of our results, they will be exploited primarily for further progress by the research groups and the wider topological dynamics community. A profound study of limit sets will contribute to our knowledge of long-term behaviour of discrete dynamical systems. It is known that rendering long-term prediction of their behaviour by computer is impossible in general, even though these systems are deterministic. Limit sets can be very complicated, this is often the case when the dynamics is chaotic. Chaotic behaviour exists in many natural dynamical systems, continuous or discrete, such as weather and climate or biological model of population growth, and chaos theory has applications in several disciplines, including meteorology, physics, computer science, economics and biology. These applications are a long-term exploitation of the fundamental research of dynamical systems.