Research in One-Dimensional and Complex Dynamics and Thermodynamical Formalism

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Abstract Jiang The goal of the first part of this project is to develop new results by combining knowledge and techniques from both Thermodynamical Formalism and Teichmuller Theory and apply them to the study of one-dimensional and complex dynamics. A generalized Fredholm Determinant Theory on the space of Zygmund functions on the circle is investigated. Notice that the space of Zygmund functions on the circle can be treated as the tangent space of the universal Teichmuller space. It is hoped that the knowledge of the spectral properties of transfer operators may be applied to the study of Teichmuller theory and vice versa. An especially interesting case is when a transfer operator is the tangent map of a non-linear map of the universal Teichmuller space. A technique called Yoccoz puzzles is also investigated as a replacement of the Markov partitions in the study of the thermodynamical properties of a non-renormalizable quadratic polynomial. In the second part of the project, the Mandelbrot set at infinitely renormalizable points and the topological properties of the Julia sets of infinitely renormalizable quadratic polynomials are investigated because these are the only remaining points to prove that the Mandelbrolt set is locally connected. A generalized Feigenbaum conjecture for folding mappings with arbitrary exponent and asymmetry is investigated in the project. The study of one-dimensional dynamics has lead to the discovery of many physical laws. One recent discovery is the universal law in chaotic phenomena: in a one-parameterized system like the family of quadratic polynomials (such a system appears in fluid and celestial mechanics, in chemistry and biology, and in social sciences), regular motions evolving into a chaotic motion can be characterized by a universal number. This new paradigm provides a rich supply of mathematical problems. In return, the study of these problems gives a deeper understanding of concepts connected with universality and develops some new tools to ma ke further progress. The goal of the first part of this project is to attack some unsolved problems and to develop new results concerned with universality. More specifically, some thermodynamical and geometric properties of one-dimensional and complex dynamical systems are investigated. The discovery of the Mandelbrot set, which was one of the direct important results from the development of a fast computer, breaks the barrier to develop some traditional geometric concepts. Traditionally, the mathematical study of geometry concentrates on smooth objects like circles, lines, and spheres. But in the real world, most geometric objects are more fractal in shape when one looks into the microscope at a geometric object. The Mandelbrot set not only breaks the visual and psychological barrier of the study of fractal geometry but also carries the universal fractal structure in a large class of systems. However, a complete understanding of the geometric structure of the Mandelbrot set is far from being finished. The goal of the second part of the project is to study the topological structure of the Mandelbrot set, and also to investigate the topological structure of the Julia set for a corresponding infinitely renormalizable point.