Back to Departmental Colloquium: Spring 2014
Departmental Colloquium
Date: Thursday, Apr 10, 2014
Time: 4:00PM
Location: LCB 219
David Damanik
Rice University
Title |
Cantor sets and Cantor measures |
Abstract |
A subset of the real line is called a Cantor set if it is compact, perfect, and nowhere dense. Cantor sets arise in many areas; in this talk we will discuss their relevance in the spectral theory of Schröodinger operators. We discuss several results showing that the spectrum of such an operator is a Cantor set, from the discovery of the first example by Moser to a genericity result by Avila, Bochi, and Damanik. A Cantor measure is a probability measure on the real line whose topological support is a Cantor set. A primary example in the spectral theory context is the density of states measure in situations where the spectrum is a Cantor set. A conjecture of Simon claims a strict inequality between the dimensions of the set and the measure for the Fibonacci potential. If time permits, we will discuss a recent result of Damanik, Gorodetski, and Yessen, which establishes this conjecture in full generality. |
Probability
Mathematical Biology