Dynamics of functions arising from Pisot and Salem polynomials

Somjate Chaiya; Aimo Hinkkanen

doi:10.1007/s11784-014-0167-2

Dynamics of functions arising from Pisot and Salem polynomials

Somjate Chaiya, Aimo Hinkkanen

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let Q(z) = zⁿP(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 (Formula presented.), 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 D such that (Formula presented.) for all z∈D.

Original language	English (US)
Pages (from-to)	371-378
Number of pages	8
Journal	Journal of Fixed Point Theory and Applications
Volume	17
Issue number	2
DOIs	https://doi.org/10.1007/s11784-014-0167-2
State	Published - Jun 23 2015

Keywords

Primary 37F10
Secondary 11R06

ASJC Scopus subject areas

Modeling and Simulation
Geometry and Topology
Applied Mathematics

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10.1007/s11784-014-0167-2

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abstract = "Let Q(z) = znP(z) − zdeg PP(z−1), where P is the minimal polynomial of a Pisot number. Boyd [Duke Math. J. 44 (1977), 315–328] showed that, for (Formula presented.), 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 Uα of α. We establish that N is a 2-fold covering map of Uα onto itself. Furthermore, there exists a conformal mapping φ of Uα onto the open unit disk D such that (Formula presented.) for all z∈D.",

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AU - Chaiya, Somjate

AU - Hinkkanen, Aimo

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N2 - Let Q(z) = znP(z) − zdeg PP(z−1), where P is the minimal polynomial of a Pisot number. Boyd [Duke Math. J. 44 (1977), 315–328] showed that, for (Formula presented.), 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 Uα of α. We establish that N is a 2-fold covering map of Uα onto itself. Furthermore, there exists a conformal mapping φ of Uα onto the open unit disk D such that (Formula presented.) for all z∈D.

AB - Let Q(z) = znP(z) − zdeg PP(z−1), where P is the minimal polynomial of a Pisot number. Boyd [Duke Math. J. 44 (1977), 315–328] showed that, for (Formula presented.), 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 Uα of α. We establish that N is a 2-fold covering map of Uα onto itself. Furthermore, there exists a conformal mapping φ of Uα onto the open unit disk D such that (Formula presented.) for all z∈D.

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Dynamics of functions arising from Pisot and Salem polynomials

Abstract

Keywords

ASJC Scopus subject areas

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