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.