Showing papers on "External ray published in 1986"
01 Jan 1986
TL;DR: In this paper, the connectivity properties of the Julia set for a polynomial have an intimate relationship with the dynamical properties of finite critical points, and the way to construct symbolic codings for the components of Julia sets for a large class of cubic polynomials is discussed.
Abstract: Publisher Summary This chapter discusses disconnected Julia sets. The connectivity properties of the Julia set for a polynomial have an intimate relationship with the dynamical properties of the finite critical points. The chapter explains the way to construct symbolic codings for the components of the Julia set for a large class of cubic polynomials. These cubics can have one critical point that iterates to infinity and another whose orbit remains bounded. The chapter discusses two filled-in Julia sets for two different types of polynomials. The black regions are the filled-in Julia set and the shading of the stable manifold of infinity corresponds to levels of the rate of escape map. In the dynamics of high-degree polynomials and in transcendental functions, there are regions on which the dynamics is determined by a low-degree polynomial.
18 citations
01 Jan 1986
TL;DR: In this paper, it was shown that algebraic integers included with all conjugates in the filled-in Julia set of a polynomial with integer coefficients are preperiodic points for the Polynomial transformation.
Abstract: We show that algebraic integers included with all conjugates in the filled-in Julia set of a polynomial with integer coefficients are preperiodic points for the polynomial transformation Therefore they must have a finite forward orbit Various extensions of this result will be discussed: polynomial with coefficients in an algebraic number ring, quantitative formulation and connection with Lehmer's problem, analogous properties for the Mandelbrot set in the parameter complex plane
1 citations