Everyone has seen movies of zooming down toward small scales in Julia sets or the Mandelbrot set; the aim of this project is to put this idea on a firm mathematical
Starting with a smooth surface embedded in 3-space, if we zoom down toward a point what we see converges to the tangent plane at that point.
But for irregular objects like fractal sets, which are nowhere differentiable, what we see never settles down. The resulting "scenery"
(the collection of limiting objects) replaces the tangent space, and this collection is acted upon by the zooming flow.
We construct and study the dynamical properties of this ``scenery flow''
for some examples: the limit set of a Fuchsian or Kleinian group, a hyperbolic Julia set. For these examples, we prove that
"Hausdorff dimension equals scenery flow entropy". This unites two
well-known dimension formulas, due to Bowen-Ruelle and to Sullivan.
As a consequence, the scenery flow of a Julia set provides an analogue of the geodesic flow of a hyperbolic 3-manifold.
Together with colleagues
IT. Bedford, M. Urbanski, P. Arnoux, and M. Talet, we have studied these and other
interesting objects, including the paths of Brownian motion, and tilings related to interval exchange transformations.