Measure preserving fractal homeomorphisms

Abstract

The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve n-dimensional Lebesque measure, where n is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged. Show more