Escape set

Escape set is a term used in the field of fractal geometry to describe a set of points in the complex plane for which a particular iterative process does not escape to infinity. The concept is primarily used in the generation of fractal art and the study of chaos theory.

Definition

In the context of fractal geometry, an escape set is defined for a function f and a point z in the complex plane. The escape set for f and z is the set of all points w in the complex plane such that the sequence w, f(w), f(f(w)), ... does not escape to infinity. In other words, the sequence remains bounded.

Applications

The concept of an escape set is used in the generation of many types of fractal art. For example, the Mandelbrot set is the escape set for the function f(w) = w² + z, where z is a complex constant and w varies over the complex plane. The Julia set is another example of a fractal that can be defined in terms of an escape set.

Escape sets are also used in the study of chaos theory. In this context, they can provide insight into the long-term behavior of dynamical systems.

See also

• Fractal
• Mandelbrot Set
• Julia Set
• Chaos Theory
• Dynamical Systems