Strange Attractors and Inverse Limit Spaces

This proposal concerns the topological structure of strange attractors of Hénon type, which emerge universally in chaotic dynamics, for example, in generic unfoldings of homoclinic bifurcations. The standard example are the Hénon attractors themselves, but in its most transparent, linearised form, they are produced by the Lozi map. Despite its widespread appearance in chaotic dynamics, the topological structure of Hénon-like attractors is poorly understood. This proposal suggests to use techniques from unimodal inverse limit spaces (UILs) and specifically classification methods that led to the solution of the Ingram conjecture, to explore the substructures and possible classification of Hénon-likeattractors. In addition, we propose to develop computer-graphical techniques to display Hénon-like attractors and their substructures on small scale. Such graphical techniques combined with methods from symbolic dynamics (two-sided kneading theory) theoretically allow one to zoom in at arbrirary scale, which should visualise substructres (both in Hénon-like attractors and UILs) that are known to exist, but whose complexity challenges the imagination. A PhD. Position is integral part of this project in specific regard of this aspect. Key advances that this project is anticipated to achieve are the following: 1. A coherent symbolic description (two-sided kneading theory) of Hénon-like maps giving a framework in which to cast the pruning front approach as well as the symbolic dynamics statements in the work on rank-1 attractors by Wang & Young. 2. A characterization of endpoints, folding points and potentially more complex substructures, such as asymptotic composants, in terms of the above symbolic description. These structures underlie the (topological and geometrical) inhomogeneities of Hénon-like attractors, and are the likely building blocks for the classification of Hénon-like attractors. 3. Computer-implemented algorithms to visualise Hénon-like attractors on arbitrary scale, so as to be a powerful tool for understanding their (sub)structures. The project will be connected to a wider research environment in continuum theory and the geometry of strange attractors, via research visits and a conference (prospectively at the Schrödinger Institute), to underline the central position of Vienna in a wider tradition of topological dynamics and continuum theory in Central Europe.

This project concerned the topological properties of so-called strange attractors, which are in reality intricate connected sets that emerge as attractors of chaotic dynamical systems. The Hénon attractor (named after the French astronomer) is a prime example, but most of the questions we posed ourselves are already of interest in the simpler setting of inverse limit spaces of interval mappings.The classification problem of such inverse limit spaces is known as the Ingram conjecture, of which one important version (the core Ingram conjecture) is only partially solved. Further research concerned the multiple ways such objects can be embedded in the plane, i.e., viewed as subsets of the plane, and which parts of the space becomes accessible.Two particular outcomes to be highlighted are:Given two tent maps with different slopes, and non-recurrent critical points, then the cores of their inverse limit spaces are fundamentally different (non-homeomorphic).For every inverse limit space of a tent map and every point x in it, there is a planar embedding making x accessible from the complement. Hence, there are uncountable many non-equivalent embeddings if the slope of the tent map lies between ?2 and 2.In addition to Prof. Henk Bruin, this project supported a PhD. student Jernej ?in?, and there were close collaborations with Dr. Sonja timac from the University of Zagreb, and especially her PhD. student Ana Anui?. The research resulted in two finished papers and one in preparation of Anui?, Bruin & ?in?. These results were frequently presented at international conferences, and will also be central in the forthcoming PhD. theses of Anui? and ?in?. In addition, two more papers were completed by Bruin & timac, and by Bobok (Prague) & Bruin.