Stereographic Metric and Dimensions of Fractals on the Sphere

Abstract

A new metric, called stereographic metric is introduced on the sphere ${{\textbf {S}}}{\setminus } N$, where N is the north pole of the sphere ${{\textbf {S}}}$. Some properties of the iterated function system and its attractor on the sphere ${{\textbf {S}}}{\setminus } N$ is characterized with respect to the stereographic metric, chordal metric and Euclidean metric. For a fixed non-empty compact subset F of ${{\textbf {S}}}{\setminus } N$, the relation between the Hausdörff dimensions of F with respect to the Euclidean metric and stereographic metric is investigated as well as a similar investigation is made for a non-empty compact subset K on the plane $z=0$ with respect to the Euclidean metric and chordal metric. We also compare the Hausdörff dimensions of a non-empty compact subset F on the sphere ${{\textbf {S}}}{\setminus } N$ and its projection $F'$ (through the stereographic map), on the plane $z=0$. Also, we found similar results for Box dimensions as well as Assouad dimensions. Finally, we define a fractal path on the sphere ${{\textbf {S}}}{\setminus } N$.