Dynamics of functions arising from Pisot and Salem polynomials

Abstract

Let Q(z) = z n P(z) − z deg P P(z −1), where P is the minimal polynomial of a Pisot number. Boyd [Duke Math. J. 44 (1977), 315–328] showed that, for \({n > {\rm deg}\,P - 2 \frac{p'(1)}{P(1)}}\), Q is the product of cyclotomic polynomials and the minimal polynomial of a Salem number, say α. In this paper, we study the dynamics of the Newton map N = z − Q/Q′ induced by Q in the immediate basin U α of α. We establish that N is a 2-fold covering map of U α onto itself. Furthermore, there exists a conformal mapping φ of U α onto the open unit disk \({\mathbb{D}}\) such that (\({\varphi \circ N \circ \varphi^{-1})(z) = z^{2}}\) for all \({z \in \mathbb{D}}\).