Connecting Elementary Cellular Automata: Topological Properties of In-between Automata

John Pedersen proposed a way of continous transformation from one elementary cellular automaton into another one. This method grants us access to the whole new class of cellular automata with wide variety of convenient local functions; we will denote this class with P. Recently it was numerically shown that automata in P may have interesting computing capabilities. We have decided to investigate Pedersen automata, using Cantor topology, in search of some properties specific to this kind of automata. We have reached two main results: there are no isometries in P and radii of ball images are non-decreasing and bounded. We also show that similar local functions of two automata do not imply small Cantor distance between their values for equal arguments.