David Damanik : The Subshift Conjecture

Posted by Irina Nenciu , part of the Departmental Colloquium.

At

April 20, 2012, 3 p.m.

In

SEO 636

Abstract

We will discuss Barry Simon's subshift conjecture, which states that the (OPUC or Schrödinger) spectrum associated with a minimal aperiodic subshift has zero Lebesgue measure. The first part of the talk will address the history of this problem, from the origins in Kohmoto-Kadanoff-Tang's 1983 paper introducing the Fibonacci Hamiltonian, through Kotani's 1989 observation that the conjecture follows from the absence of non-uniform hyperbolicity, to proofs of certain kinds of uniformity in certain settings by Furman in 1997 and Damanik-Lenz in 2006. In the second part we will present recent joint work with Artur Avila and Zhenghe Zhang that produces a counterexample to the conjecture.