i am a bit surprised with tom's line of reasoning: "it's not topology, so it shouldn't be mentioned"
I don't know what I wrote suggesting that I think that; I don't. In two of the papers advertised (first paragraph of the first paper and bottom of p.3 of the second), I listed some other types of self-similarity that I'd like to investigate. Let me expand on the possibilities. 1, 2. Set-theoretic and topological: these are the types of self-similarity that I understand by far the best at present. 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). For instance, given a polynomial p(x) in N[x], you can consider solutions to x = p(x) (in rings, rigs, distributive categories, and rig categories): see the work of Robbie Gates, Peter Hines (TAC vol 6), Marcelo Fiore and I, and probably many others. 4. Conformal/analytic: this is the natural setting for discussing the self-similarity of Julia sets of complex rational functions. 5. Metric: e.g. the Koch snowflake is topologically just a circle, but has interesting metric self-similarity. 6. Measure-theoretic: cf. some of Peter's postings of 1999-2000 concerning integration. 7. Order-theoretic: both Peter's and Dusko's/Vaughan's characterizations of a real interval produce its ordering. (I have some idea of how to handle order in the much more general situation discussed in my papers, but it's early days.) 8. Categorical: e.g. the category of strict omega-categories is the terminal coalgebra for the endofunctor of CAT defined by V |--> V-Cat. (This was an idea of Carlos Simpson.) The same goes for globular sets (omega-graphs), changing V-Cat to V-Graph. 9. Statistical: there are so-called random fractals - e.g. take a black square; divide it into a 3x3 grid and white out each of the 9 subsquares with probability p; do the same to each black subsquare; continue ad infinitum. 10. Algebraic: the Thompson groups are in some sense highly self-similar. (These groups may be best known to readers of this list from Freyd and Heller's work on homotopy idempotents, but are also very tree-y in nature.) Here's a question belonging to (10), to which I don't know the answer. Let C be the category whose objects are triples (V, v_0, v_1) where V is a vector space and v_0 and v_1 are linearly independent vectors in V, and whose maps preserve linear structure and the `basepoints'. There's a `wedge' functor C x C --> C defined by (V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1) where (V + W)/~ is the direct sum with v_1 identified with w_0. (So dim(V wedge W) = dim V + dim W - 1.) There's then an endofunctor G of C given by self-wedging. Question: what, if any, is the terminal G-coalgebra? (I suspect the answer is something to do with measure/integration - again see Peter's previous postings - but really have no idea.) Finally, re citations: I'll stick in a Pavlovic-Pratt reference, as suggested. All the best, Tom 26-Nov-2004 13:34:46 -0400,1447;000000000000-00000000