A study of fractals based on non commutative methods
Seminar Room 1, Newton Institute
A fractal set in a Euclidian space is by nature non smooth, and the concept of a differentiable function on it makes no sense. The methods from noncommutative geometry are designed to describe smooth structures in a language based on operators on Hilbert spaces, modules over self-adjoint algebras and cohomological invariants. This language may also be used in the case where the algebra is a subalgebra of the continuous functions on a compact fractal set.
Connes' book "Noncommutative Geometry" contains several results along this line. One of the spectral triples Connes constructs is a a countable sum two-dimensional modules. We have made a general study of this in the general setting of a compact metric space, and obtained some results of general nature.
We have then looked at Sierpinski's Gasket, and it is clear that sums of two dimensional modules is not the right sort of module here, the reason being that such a module can not detect all the holes in the gasket. We have then constructed a spectral triple for the Sierpinski Gasket, based on all the holes, i. e. an infinite sum of modules based on all the triangles in the gasket. It turns out that this spectral triple measures the geodesic distance on the gasket, it gives the Minkowski dimension of the gasket, it induces a multiple of the self-similar Hausdorff measure via the Dixmier trace and it detects the holes, by a non trivial pairing with the K-theory.