The radius of convergence (Julia set
) is an important characteristic of the corresponding basin of attraction.
(a) The Julia set
of f has an empty interior if the Fatou set is non-empty.
Since [+ or -] 1 are attractive fixed points of B, so the Julia set
](B) is the boundary of the basins of attraction S(-1) and S(1); that is,
A point [z.sub.0] is in the Julia set
if and only if dynamics in a neighborhood of [z.sub.0] shows strong dependence on the starting conditions, so that nearby initial conditions yield to wildly different behavior after a number of full iterations.
A complex number z (or a + ib) is a member of a Julia set
, for a given real number c, if z is the first member of a sequence, defined by an iterative function, which remains bounded.
Fig ure 1.3 shows the initial part of the evolution in the case of the usual Bessel functions (for which the measure [micro] is absolutely continuous), and Figure 1.4 displays the same information in the case of a singular continuous Julia set
The Mandelbrot set is closely related to the Julia set
. Namely, if the value of the coefficient is inside the main cardioid of the Mandelbrot set, Julia set
will represent the misshapen circle.
The complement of F(f) in C is called the Julia set
J(f) of f.
With the support of her family, friends and boyfriend, Julia set
about rebuilding her life and the confidence which played such a big part in it.
Orbits and the Julia set
. Australian Senior Mathematics Journal, 16 (2), 53-54.
Armed initially with nothing but a good idea, Julia set
off on a path which has since seen her embark on a cheese-making course with the UK's experts, as well as converting the family garage into a custom-made workspace complete with a cheese vat.
up her business in September with the help of a Business Start-up Grant from South Tyneside Means Business.