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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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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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  • $\begingroup$ offhand: Has anyone restated the scenery flow material in the language of sheaves? $\endgroup$ – graveolensa Jun 17 '13 at 17:47

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