Almost periodic Jacobi matrices associated with Julia sets for polynomials

Abstract

Letℐ be the Jacobi matrix associated with polynomialT(z) of degreeN≧2. The spectrum ofℐ is the Julia set associated withT(z) which in many cases is a Cantor set. Letℐ (1) denote the result of omitting the first row and column ofJ. Then it is shown that the spectrum ofℐ (1) may be purely discrete. It is also shown that forT(z)=α NCN(z/α) for α> $$\sqrt {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} $$ , whereC N is a Chebychev polynomial the coefficients ofℐ andℐ (1) are limit periodic extending the work of Bellissard, Bessis, and Moussa (Phys. Rev. Lett.49, 701–704 (1982)).