Fractals, flows and randomness

We study fractal-like geometric objects by means of the flow defined by zooming toward a point of an ambient Euclidean space. This ``scenery flow" provides an analogue for the geodesic flow associated to a Kleinian group.

One consequence is a dimension formula for hyperbolic Julia sets which unites and simplifies the Sullivan and Bowen-Ruelle formulas to: Hausdorff dimension equals scenery flow entropy.

For fractal sets, the translation scenery flow has a natural conservative ergodic infinite measure. This observation builds a bridge between fractal geometry and the probability theory of recurrent events, suggesting on the one hand new theorems for the Fuchsian case and on the other a new interpretation of some results on countable state Markov chains due to Feller and Chung-Erdös. Interesting examples are seen in the intermittent return-time behavior of maps of the interval with an indifferent fixed point.