The Mandelbrot set can be defined as the set of points c in the complex plane that do not go to infinity when iterating the function z_n+1 = z_n² + c, for z₀ = 0. The iteration defines an orbit. Points in the Mandelbrot set can be of various types:

1. outside the set and no where near it: the orbit diverges to infinity;

2. inside the set and not on the boundary: the orbit converges to a finite attracting point or cycle;

3. at the edge of the set and on the tip or branch of a filament: the orbit converges to a finite repelling cycle;

4. at the edge of the set and on other filament points which present irrational external angle: the orbit is chaotic;

5. at the edge of the set and on the cusp of a cardiad or bud attachment point (one or more rational external angles): the orbit is parabolic and goes to a weakly attracting cycle or point.

The Mandelbrot set does not present true self-similarity. The orbits are unbounded for |z| > 2. The boundary of the Mandelbrot set is a very complicated fractal with a fractal dimension of 2. Bounded orbits may attract to a fixed point, aperiodic cycles or they may be chaotic.

In turn, for each c in C a Julia set is defined by the function Q_c=z²+c. Each set corresponds to the boundary of the set of points z whose orbits escape to infinity. A known theorem characterizes such points in the real plane:
Theorem: Consider f(x) = x²+c with real c. An orbit escapes to infinity if and only if the starting point z₀ is greater than 1+c or lower than -1+c.

For the complex plane, such a characterization is not that simple:
Theorem: Let f(z) = z² + c where |c| < 2. Consider the iteration z_n+1 = z_n² + c. If any point of the orbit generated by z₀ has absolute value greater than 2, the orbit escapes to infinity.

Moreover the topology of the Julia set Q_c depends on the orbit of 0:
Theorem (Fundamental Dichotomy): Let c belong to the complex plane. If the orbit of 0 for Q_c escapes to infinity, then the Julia set is topologically identical to the Ternary Cantor Set, i.e., it is infinitely disconnected. If the orbit of 0 for Q_c is limited, then the Julia set is connected.

The Mandelbrot and Julia sets are related. Actually, the Mandelbrot set indicates when a certain Julia set is connected, since the Mandelbrot set is the set of points c such that the orbit of 0 is limited. This is why the Mandelbrot set can be described as an atlas that maps the behaviour of a Julia set. While the Mandelbrot set is always connected, the Julia set is connected if and only if it is associated to an inner point of the Mandelbrot set.

Each Julia set is associated to the point in the Mandelbrot set marked with the blue cross.