Abstract
In this paper, we discuss Hausdorff measure and Hausdorff dimension. We also discuss iterated function systems (IFS) of the generalized Cantor sets and higher dimensional fractals such as the square fractal, the Menger sponge and the Sierpinski tetrahedron and show the Hausdorff measures and Hausdorff dimensions of the invariant sets for IFS of these fractals.
Highlights
Any fractal has some infinitely repeating pattern
We study the Cantor set and formulate iterated function system with probabilities of the generalized Cantor sets and show their invariant measures using Markov operator and Barnsley-Hutchison multifunction [2]
Let S be an invariant set for iterated function systems (IFS) of the square fractal which is represented by the following: w1 ( x, y)
Summary
Any fractal has some infinitely repeating pattern. When crating such fractal, repetition of a certain series of steps is necessary which create that pattern. Iterated Function System is another way of generating fractals. We discuss Hausdorff measures and Hausdorff dimensions of the invariant sets for iterated function systems of these fractals. We can define iterated function system as follows: Let 0 < β < 1. We present Hausdorff dimension of the invariant set for contracting maps. Hausdorff measures and Hausdorff dimensions of the generalized cantor sets.
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