Cantor Spectrum for the Almost Mathieu Equation

Recently, there has been an explosion of interest in the study of Schrijdinger operators and Jacobi matrices with almost periodic potential (see, e.g., the review [ 161). The general belief is that generically the spectrum is a Cantor set, i.e., a nowhere dense perfect, closed set. Since it is easy to prove that the spectrum is closed and perfect (see, e.g., [2]), the key is to prove that the spectrum is nowhere dense. This is definitely not always true: There are very special finite gap potentials, i.e., V’s for which -d’/du* + V(X) has a spectrum which is the finite union of closed intervals [6]. For the special class of limit periodic potentials, Chulaevsky [5], Moser [ 131 and Avron and Simon [2] have proven that generically (in the sense of dense G6) the spectrum is nowhere dense. Our goal in this paper is to prove that the spectrum is nowhere dense for another special class of potentials, specifically for the operators on I,,