Fractals, Lyapunov exponents, and spectral theory of almost Mathieu operator (AMO) of Jacobi type
Abstract
Fractals are used to model many phenomenons in diverse areas of sciences such as chemistry, ecology, material sciences, and medicine. They also appear in many areas of mathematics such as dynamical systems, partial differential equations, and operator theory. Because of their use in such a variety of fields, a deep and far-reaching intrinsic theory of calculus has been developed on a class of fractals. This theory now provides the mathematical foundation needed to rigorously modeled phenomenons on fractals. In this context, analysis on fractals has emerged during the last three decades with the introduction of a Laplacian on certain fractals. For instance, extensive research in this area has resulted in a better understanding of Fourier series and certain (Laplacian based) differential equations on fractals. From a different perspective, certain fractal sets have been associated with the spectra of some differential operators. For example, the Hofstadter butterfly is known to be related to the spectrum of certain 1-dimensional Schroedinger operators. In this context the Lyapunov exponents (LEs) appear to be important tools that link the dynamics of these operators to their spectral properties. In this proposal we seek to use tools from analysis on fractals to investigate spectral problems of almost Mathieu operator (AMO) of Jacobi type, and the corresponding LEs.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 02, 2019
- Source ID
- W911NF1910366
Entities
People
- Kasso Akochayé Okoudjou
Organizations
- Army Contracting Command
- United States Army
- University of Maryland