Division of Information Technology, Engineering and the Environment
School of Mathematics and Statistics
School of Mathematics and Statistics
In this talk, we explore a family of polynomials whose roots are related to the Mandelbrot set. Indeed, these roots correspond to some $k$-periodic points of the iteration defining the Mandelbrot set. We discuss some of a variety of approaches to compute the roots of these polynomials; classical iterative schemes, eigenvalues of companion matrices and a novel family of recursively defined matrices. Time permitting, we look at an experimentally-discovered asymptotic series for the largest-magnitude roots.
Joint work with Piers W. Lawrence and David J. Jeffrey.
The uniform classification of infinite-dimensional spheres, developed in relation with the solution of the distortion problem is more balanced than the continuous, isometric, Lipschitz or uniform classifications of infinite-dimensional Banach spaces. It allows to transfer a group structure, group actions and other metric-related constructions from one space onto another.
We show that the uniformly continuous homeomorphisms can be "upgraded" to the Hölder ones in the classical setting and establish the explicit and, occasionally, sharp exponents of the Hölder regularity for pairs of concrete spaces, including various Besov, Lizorkin-Triebel, Sobolev, sequence, Schatten-von Neumann and other Banach spaces (including lattices and more general non-commutative spaces).
Our function spaces are allowed to be anisotropic and can be defined in terms of differences, local approximation by polynomials, the coefficients of wavelet expansions, Littlewood-Paley decompositions or a functional calculus. Not every equivalent norm is geometrically friendly.
These results appear to have close ties with the presence of a remarkable phenomenon from the infinite-dimensional approximation theory discovered by Tsar'kov for the uniform mappings between pairs of Lebesgue spaces and the problems of extension and interpolation of the mappings between the pairs of the spaces under consideration. Our results show the way of substituting the qualitative global stability with the quantitative one.
Among the applications of the main results are multiple examples of spaces that do not allow any C*-algebra structure but can be endowed with a homogeneous Hölder group structure.
