The Open Set Condition and Neighbor Maps in Fractal Geometry

Abstract

Neighbor maps are a new development in fractal geometry that can be used to determine if an iterated function system (IFS) obeys the open set condition (OSC). Neighbor maps also describe the topology of the fractal attractor. In this thesis, the de nition of the set of proper neighbor maps is generalised to any IFS comprising contractive similitudes. It is proven, an IFS of similitudes obeys the OSC if and only if the identity map is not in the closure of the set of proper neighbor maps. The extended de nition of the set of proper neighbor maps is used to calculate several neighbor graphs of the generalised Sierpinski triangles. It is then proven, if a generalised Sierpinski triangle has scaling factors that obey the algebraic condition, then its neighbor graph is of nite type. The converse is proven to only hold for the Sierpinski triangle and the Steemson triangle. Neighbor maps are also applied to fractal tiling theory to discuss the prototile set. Show more