Topology and Low-Dimensional Dynamics

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The project is concerned with complex, plane, and real dynamics. The overarching theme of its complex dynamical part is using Thurston laminations to describe the combinatorial model for the boundary of the cubic connectedness locus, to extend the Fatou-Shishikura inequality, and to construct the finest topological/combinatorial dynamical models for Julia sets. In the part on plane dynamics the project seeks to replace holomorphic tools from complex dynamics with tools based upon the expansiveness of maps and use this idea to study plane maps called "expanding polymodials." In particular, the project will investigate "c-tent" plane maps that combine the expansive properties of tent maps with properties of complex quadratic maps. As such, they should play the same role for quadratic complex maps that tent maps play on the interval. In the realm of one-dimensional dynamics, the project pursues research in two directions: (1) the principal investigator wants to determine all points on the interval such that small perturbations around them change the dynamics of the map (this serves as a pointwise version of stability, is new even for unimodal maps, and requires original tools); (2) the project will develop further the rotation theory for interval maps, complex quadratic maps, and billiards.This project will use combinatorial methods to study dynamical systems and their related sets. An appropriate model of an intricate planar set (e.g., a fractal set) should shed light upon its structure, allowing one to predict its properties. The research will also investigate the connection between critical phenomena in dynamics (normally easy to observe) and long-term phenomena that are not so obvious (such as convergence to periodic behavior in the future). Thus, these two topics aim at predicting properties of sets and dynamical systems, information that is necessary for understanding many concrete physical phenomena. In addition, the connection between various types of planar dynamical systems will be explored, such as systems that are locally compositions of shrinking/expansion and rotation or systems that locally expand the distance between points. The principal investigator will also study points such that small perturbations around them drastically change the underlying system, as well as periodic points at which a system exhibits rotational properties, as part of a more complicated scheme than a rotational dynamical system. All these topics require new techniques and can be used as a basis for theses, and thus for the development of human resources.