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I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are connected. I have come up with 3 classifications so far and think that all self similar sets fit into these categories although I have not been able to prove this fact and am looking for some help proving it or a counterexample in another category (most likely non rectifiably connected sets that are not Jordan arcs). I shall state the categories and give an example from each to give some clarification.

Non rectifiably connected Jordan arcs (Koch arc)

Rectifiably connected sets (Sierpenski Gasket)

Totally disconnected sets (C x C : a 2 dimensional cantor set)

Thank you for your help. I apologise if I have been unclear anywhere, I will try and clarify any points that need it.

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    $\begingroup$ I am given to understand that such open ended questions are not encouraged on this site. To focus it you should add more information about what you intend to use this classification for and what kinds of contractions you are interested in. That said, where would you put as set that is a Cantor set cross an interval? It is not connected but also not totally disconnected. $\endgroup$
    – BSteinhurst
    Commented Oct 13, 2011 at 3:30
  • $\begingroup$ I am not familiar with the adjective "rectifiably" so I cannot tell if your classification includes tree-like (e.g., fern pictures). You should also note that CxC is homeomorphic to C. $\endgroup$
    – Matt Brin
    Commented Jan 17, 2012 at 15:39

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