Rigidity, Universality, and Complexity in Dynamics

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The main theme of my research is the study of complex behaviour generated by simple dynamics. One of the key themes in Dynamics is the study of universal laws which emerge from the complex behaviour of dynamical systems. Such universalities are well-known in physics. They can be understood as a self-organization mechanism, which arises from chaotic behaviour. Perhaps the most famous examples of them are Feigenbaum-type universalities in one-dimensional dynamical systems, which have been intensively studied in the last three decades. These studies founded a new field, known as renormalization theory, which revolutionized the study of dynamics. In a series of papers I completely resolved one of the two main cases of one-dimensional universalities (the Lanford's universality for critical circle maps). Closely related to universality is the phenomenon of rigidity. It postulates that dynamical systems which exhibit the same universal properties are related by a smooth change of coordinates. My proposal outlines a program of study of some of the central questions of universality and rigidity in one-dimensional dynamics. It is a well-known fact that a physical system guided by simple dynamical laws can be difficult to model and predict. A famous example of this is the Lorenz system, which describes a very simplified weather model. Even tiny differences in the initial conditions of the model may lead to a widely diverging weather predictions (a so-called butterfly effect). Nevertheless, it is generally believed that while individual predictions may differ, the overall picture of the dynamics remains the same, and may be simulated numerically. In the Lorenz example, the global weather patterns are described by the Lorenz butterfly attractor, which is a well-defined mathematical object. It is easily computed (indeed, the picture has become famous). In a series of recent works, we have tested the intuitive assumption that the attractor of a simple dynamical system can be modeled on a computer. Surprisingly, for an archetypical family, known as Julia sets, this is generally false. This work opens a chapter of research which is important for practitioners, as well as for theoretical dynamicists and computer scientists. My proposal lays out a plan of further study of these challenging questions.