  The structure of a self-similar set with open set condition does not change
under magnification. For self-affine sets the situation is completely
different. We consider planar self-affine Cantor sets E of the type studied by
Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the
horizontal axis is an interval. We show that within small square neighborhoods
of almost each point x in E, with respect to many product measures on address
space, E is well approximated by product sets of an interval and a Cantor set.
Even though E is totally disconnected, the limit sets have the product
structure with interval fibres, reminiscent to the view of attractors of
chaotic differentiable dynamical systems.
