Special Dynamical Systems Seminar
Albert M. Fisher
University of Sao Paulo
Fractals, flows and randomness
We study fractallike 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 BowenRuelle 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
ChungErdös. Interesting examples are seen in the intermittent
returntime behavior of maps of the interval with an indifferent
fixed point.
