Special Dynamical Systems Seminar
Albert M. Fisher
University of Sao Paulo
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