Symmetry groups of fractals

This paper contains computer pictures of generalised Mandelbrot and Mandelbar sets, and their associated Julia sets, from which it is evident that their symmetry groups possess an elegant and simple structure. We show that (i) the Mandelbrot set M(p) generated by the iteration zt+ ~ = ztp + c remains invariant under the symmetry transforms of the dihedral group Dp_ 1 (i.e., these are isomet-ries of M(p)); (ii) the Mandelbar set M(p) is invariant under the isometries inDp + 1; and (iii) the Julia sets of points inside M(p) (or M(p)) are invariant under the isometries in either Dp or just the cyclic group Cp, depending on whether the seed point is on or off a symmetry axis of the parent Mandelbrot (or Mandelbar) set. The proofs are relatively easy, but showing that there are no other isometries of these sets is not so straightforward. As is often the case in the theory of chaos, what is obvious geometrically is difficult to prove analytically. For the generalised Mandelbrot and Mandelbar sets with even p we have in fact proved that the dihedral symmetry transforms are the only isometries of these sets, but the method does not appear to be applicable to odd p, or to the Julia sets.