ABSTRACT
Fractals are geometric objects that can be subdivided into parts, each of which is a copy of the whole. Definitions and estimation methods of fractal dimension and invariant measures for chaotic systems, as well as some examples of well-known fractals and chaotic maps are also given. Fractal dimension is a statistical quantity that gives a measure of how closely a fractal fills space as it is scaled down. In general, the dimension of a system can be thought of as the number of degrees of freedom, or the number of independent variables needed to define the state of the system. The Sierpinski triangle, named after the Polish mathematician Waclaw Sierpinski, can be constructed in a two-dimensional plane starting with an equilateral triangle with the base parallel to the horizontal axis, and successively removing the middle fourth of the triangle, thus leaving a triangular hole inside the original triangle.
