Fractal lake

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In geometry, and less formally, in most Fractal art software,^{[citation needed]} the fractal lake of an 'orbits' (or escape-time) fractal, is the part of the complex plane for which the orbit (a sequence of of complex numbers) that is generated by iterating a given function does not "escape" from the unit circle.^{[citation needed]} The lake may be connected or disjoint, and it may also have zero area.

Orbits that are initialized inside the lake are either eventually captured by zero, captured by another point inside the unit circle, or may oscillate through a set of finite values indefinitely without ever converging to a fixed point. These points are described as being Inside the lake. Inside points are often detected for the purposes of using a different coloring method, in fractal rendering software^[1]

By this definition, the points of the Mandelbrot set form a "fractal lake", which is why the Mandelbrot set is also sometimes known as the "Mandelbrot Lake"^[2], or the "lake of the Mandelbrot Fractal".

Many complex valued functions with an attractor at the origin define a fractal when this aspect of their orbits' behavior is categorized. Some of the orbits are attracted to the origin; some are periodic; some are attracted to other attractors, including possibly an attractor at infinity.

For a given function there is a Julia fractal for each point on the complex plane. The Julia sets that correspond to points inside the Mandelbrot set are connected; those that correspond to points outside of the Mandelbrot set are disconnected.