Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

The proposed work involves a broad range of problems in the theory of dynamical systems (in particular, hyperbolic theory) and its applications to and relations with dimension theory, mathematical physics (including coupled map lattices), ergodic theory, and statistical mechanics. Projects in the dimension theory of dynamical systems include: the description of several multifractal spectra (including dimension spectra, entropy spectra, and spectra for Lyapunov exponents) and the corresponding multifractal decompositions for various classes of dynamical systems of hyperbolic type (non-conformal expanding maps, Axiom A diffeomorphisms, etc.); and the establishment of the multifractal rigidity phenomenon for multidimensional conformal expanding maps (the classification of dynamical systems up to multifractal spectra). In the theory of coupled map lattices (CML), the principal investigator will study: infinite-dimensional SRB measures for CML, which includes establishing the characteristic property and thermodynamical limit, describing the Lyapunov spectrum, and proving the entropy formula; and the stability of traveling wave solutions of CML and relations between traveling wave solutions for CML and the corresponding PDE (such as Ginzburg-Landau equation, Kolmogorov-Petrovski-Peskunov equation, Huxley equation, etc.). The principles of symmetry and self-similarity are nature's most beautiful creations. For example, they are expressed in fractals which are famous for their beautiful but complicated geometric shapes. Examples of fractals vary from well-known ones-cost lines or mountain ranges-to less known-distribution of stars in galaxies and galaxies in the universe or root systems of plants. Dimension theory is a mathematical theory which is designed to explain fractals' structure. And in dynamics the presence of invariant fractals often results in unstable ``turbulent-like'' motions and is associated with ``chaotic'' behavior. Thus the study of fractals can help understand the most complicat ed phenomena such as turbulence in the ocean or atmosphere. The proposed work involves research in a recently developing area which lies in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, the principal investigator intends to provide a comprehensive and systematic treatment of modern dimension theory in dynamical systems. Results are expected to be of great importance not only to advanced mathematicians but to a wide range of scientists who depend upon mathematical modeling of dynamical processes, including physicists, specialists in numerical modeling, engineers, molecular biologists, etc.