There is one important aspect of self-similarity, at least as it is understood in the context of:
4. Conformal/analytic: this is the natural setting for discussing the self-similarity of Julia sets of complex rational functions.
3. Type-theoretic: recursive datatypes can be understood as self-similar objects, the best-known example being trees (which can be characterized as a final coalgebra).
that seems to have been 'lost', or at least been made obscure enough that I could not see it anymore. The reason that proving self-similarity of some (conformal/analytic) fractals is quite difficult is because the definitions of self-similarity used always insist on 'bounded distortion', in other words you are allowed to diform the whole before re-injecting it as a part, but the distortion has to be bounded. For iterated function systems, since all the transformations are linear, this is trivial to show. But for Julia sets, since the 'natural' self-similarity involves non-linear transformations, proving bounded distortion is much more difficult. The 'puzzle pieces' of Yoccoz were invented explicitly to provide a tool for showing bounded distortion. One can show that most Julia sets are self-similar away for the orbits of critical points; for critical points embedded in the Julia set, there are known dynamical conditions which imply bounded-distortion, and then self-similarity. But there are definitely still some open cases. For some settings, like this bounded distortion is obvious, since there is *no* distortion at all. Did I just 'miss' some condition that would insure bounded distortion? [I admit to have only read Leinster's 'overview' paper, the other 2 papers are on my to-read-late pile]. Jacques 26-Nov-2004 19:47:23 -0400,1336;000000000001-00000000