Julia Sets and Autonomous Differential Equations

Abstract

Publisher Summary This chapter discusses the theory of moments of balanced measures on Julia sets and explains its connection with the study of certain autonomous ordinary differential equations. The chapter emphasizes on how, starting from a rational mapping F(z) of the complex plane into itself , under the appropriate conditions, one can construct an associated self-adjoint operator whose spectrum is a fractal related to F(z)—namely, its Julia sets. The chapter explains how simplified Hamiltonians for complicated physical systems with fractal structure might be developed. To illustrate how a fractal can be associated with an iterated rational map, the chapter considers Newton's method applied to find the zeros in the complex plane ℂ of f (z) = z 3 -z.