Examination of codings and intirinsic metrics on self-similar sets

Fractals are one of the most popular research topics of recent times because of their relationship with the nature. One common feature of all fractals is self-similarity. Self-similar sets can be studied in different classes; strong self-similar sets, weak self-similar sets and randomly self-similar sets. Some classical models of strong self-similar sets are Cantor set, Cantor dust, Sierpinski triangle, Sierpinski carpet, Koch curve, Box fractal, Sierpinski tetrahedron and Menger sponge. The Sierpinski propeller and adjecent Sierpinski triangle can be given as examples of weak sel-similar sets.Julia sets and Mandelbrot set are some examples of randomly self-similar sets. Self-similar sets have different codings and the intrinsic metrics that are compatible these codings due to their structures. In this thesis, firstly the codings of classical fractals will be investigated. After examining the intrinsic metric formula defined on the code set of Sierpinski gasket in the literature, the intrinsic metric formulas will be constructed on the Sierpinski propeller and adjecent Sierpinski triangle which are some example of weak self-similar set. Finally, by using the intrinsic metric formula on the added Sierpinski triangle the points which have two or more shortest paths will be classified. The points which have for n=0,1,2,3,... the number of geodesics 2^n,3.2^n+n and infinity have shown and some code representations of these points have been expressed according to the number of geodesics.