Topology of Lorenz-like attractors

This project entitled ``Topology of Lorenz-like attractors`` carried out by Jernej Cinc aims to study the three dimensional paradigm of chaos called Lorenz attractor and its generalizations called Lorenz-like attractors. In the early 1960`s astronomer and physicist Edward N. Lorenz studied a model for heat convection and gave a simplified model to study using a system of three differential equations. Solutions to these equations became known as Lorenz attractors. Despite the fact that these attractors have been known to mathematicians for around 40 years, the topology (fine structure) of the attractors has not been studied in details. The main obstacle that such a study has not been established yet was the lack of techniques necessary to perform such a study. The proposed project suggests to develop such techniques arising from well established symbolic dynamics on interval maps and perform an analogous study as it was given on a related field called inverse limit spaces of tent maps. After the symbolic setup is given, the project suggests to harvest the results that are offered by the organized symbolic setup. Namely, the second goal of the project is to characterize topological inhomogeneities (i.e. topological features that are used to distinguish among attractors) that arise in the Lorenz attractors. Locally, Lorenz attractors are often simple. However, we conjecture that this is not the topologically typical case. We provide this conjecture and its consequences based on the knowledge developed during the study of tent inverse limit spaces initiated by Bruin and Barge, Brucks and Diamond respectively in the late 1990`s. Once we have a good knowledge about the topological inhomogenaities that appear in the Lorenz-like attractors, the aim of the project is to give a classification of the Lorenz-like attractors. We will introduce modifications of techniques that were developed by Barge, Bruin and Štimac in 2013 in order to prove the long-outstanding classification problem about the tent inverse limit spaces called the Ingram conjecture. The last problem of the project is to give a characterization of possible self-homeomorphisms of Lorenz-like attractors and study their properties. Analogous study in the setting of tent inverse limit spaces was given recently by Bruin and Štimac and it followed implicitly from the techniques used for solving the Ingram conjecture. We conjecture that a characterization of possible self-homeomorphisms of Lorenz attractors will be consequent to the topological classification in this setting as well. The findings of this project could open an avenue of mathematical research since we intend to introduce already tested techniques to an area of mathematics that has not exploited these techniques yet.

This project was concerned about topological, dynamical and measure-theoretic properties of the so-called ``strange attractors'', which are intricate connected sets that emerge as attractors of chaotic dynamical systems. One of the prime and most widely recognised strange attractors are the Lorenz attractors. In particular, due to the observations on Lorenz attractors the widely used term "butterfly effect" was coined. In this project we posed ourselves questions related to Lorenz attractors as well as other one-dimensional compact connected sets that appear as strange attractors. The first main result (proved jointly with Piotr Oprocha (AGH Krakow)) determined that the pseudo-arc, object of much interest in Continuum Theory, connects Measure and Continuum Theory. Namely, we showed that minimal invertible extensions of generic chaotic interval functions live on the pseudo-arcs. Furthermore, we studied attractors induced by these generic functions and showed that they posses several good measure-theoretic and statistical properties and can be well modelled by computer simulations. The second main result of this project is the resolution of a more than 30 years old problem, the entropy conjecture of M. Barge from 1989 which was done jointly with Jan Boroński (AGH Krakow) and Piotr Oprocha (AGH Krakow). The conjecture stated that the pseudo-arc admits homeomorphisms with arbitrary non-negative real value of topological entropy and we answered the conjecture in the positive. The final main result of this project was establishing a theory to study local topological properties of strange attractors which was done jointly with Ana Anušić (University of São Paulo) et al.. In a series of papers we developed novel tools useful to detect and distinguish different types of inhomogeneities in one-dimensional strange attractors.