The project proposes a study of topological properties of strange planar attractors, for simplicity focusing mainly on Lozi attractors. The family of Lozi maps is a family of planar homeomorphisms which depends on two real parameters a, b and is given by L_(a,b)(x,y)=(1+y-a|x|, bx) for every (x,y) in the real plane. The attractor of L_(a,b) is the set which attracts all the points of its neighbourhood. Topologically, such set is a continuum, i.e. a compact and connected metric spaces. It is known that for a large set of parameters such an attractor is indecomposable and the action on it is topologically transitive. However, not much else is known about the topological structure of such spaces. Intuitively they resemble (but are unfortunately not homeomorphic to) the family of unimodal inverse limit spaces. That family has generated a vast amount of research during the last 30 years, predominantly chasing after the proof of the Ingram conjecture, stating that all the non-trivial inverse limits generated by a logistic bonding map (the family of logistic maps is a distinguished family of unimodal interval maps; it is a full family) are mutually non-homeomorphic. The Ingram conjecture was proven (in positive) in 2012 by Barge, Bruin and `timac. Dynamics of every unimodal map has a very nice symbolic description which generalizes to the inverse limit spaces. Every point in the space can be described as a two-sided infinite sequence of two symbols and the longer the symbolic representations of two points agree, the closer they are. Moreover, the action of the shift homeomorphism acts as the shift on the symbol space. That makes the study of unimodal inverse limits and their natural homeomorphisms easily approachable. For example, one can symbolically describe arc-components of the space, or the points which locally fail to resemble the Cantor set of arcs (called folding points). One can also distinguish a set of endpoints within the set of folding points. Moreover, it is possible to symbolically describe the set of accessible points and the prime end structure once the space is embedded in the plane.We know that Lozi family possesses some phenomena which makes its attractor substantially different than any unimodal inverse limit space. However, every such attractor is a one-dimensional continuum with no simple closed curves, and thus it is a tree-like continuum. That means that it is possible to approximate the space by finer and finer nice open covers whose nerves are trees. One of the proposed questions in this project is to find such covers and describe the space as a tree-like continuum. Moreover, it would be nice to find a single space and a single bonding map in such construction such that the dynamics on the attractor is given for free via the shift homeomorphism. Such a space might be more complicated than a tree, but can possibly be a certain dendrite. Although Lozi attractors are not necessarily as simple as unimodal inverse limits, we know that every Lozi map can be realized as a pruned horseshoe. That gives a certain symbolic description of its attractor. Loosely, as in the unimodal inverse limit case, every point will be described by a sequence on two symbols, with certain admissibility conditions. Another proposed question of this project is to make that description operable as in the unimodal inverse limit space, and to use it to describe the arc-components, folding points, endpoints, accessible sets, and the prime end structure of Lozi attractors.
News published in Agência FAPESP Newsletter about the scholarship: