Eigenvalues in Filled Julia Sets

Abstract

SummaryWe show how Julia sets can be introduced very naturally in a junior-level linear algebra course, as a way of exposing students to the contemporary area of complex dynamics. The standard definition of the filled Julia set of a polynomial is generalized to the setting of polynomial iteration of matrices. We prove that the eigenvalues of any matrix bounded under iteration by a polynomial must lie in the corresponding filled Julia set. A partial converse is obtained if the matrix is assumed to be diagonalizable. Still another partial converse is proven by assuming the spectrum of the matrix is contained in the interior of corresponding filled Julia set.