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In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not find any material online about this. Does anyone have reference on that?

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Cross-posted (under the name Rob): math.stackexchange.com/questions/422449/… –  Benjamin Dickman Jun 17 '13 at 11:22

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up vote 5 down vote accepted

Fractals and trees can be connected through the machinery of CP-processes, which are a kind of discrete-time version of the scenery flow. This has been studied intensively in the last few years by my colleague Pablo Shmerkin and coauthors: I think that the paper "Local entropy averages and projections of fractal measures" by Hochman and Shmerkin should contain enough references to help you find what you're interested in.

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offhand: Has anyone restated the scenery flow material in the language of sheaves? –  deoxygerbe Jun 17 '13 at 17:47

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