Dirichlet forms on non self-similar sets : Hanoi attractors and the Sierpiński gasket

In this thesis we study a class of non self-similar fractals, the so-called Hanoi attractors of parameter alpha. We investigate the geometric and analytic relationships between the Hanoi attractors and the Sierpinski gasket, which is one of the most studied self-similar fractals.
The first part of the thesis treats the problem from a geometric point of view: For each alpha in (0, 1/3) we construct the Hanoi attractor and prove that the sequence of attractors converges to the Sierpinski gasket in the Hausdorff metric as alpha tends to zero. Moreover, we prove convergence of the Hausdorff dimension as alpha tends to zero.
The second part of the thesis deals with the construction of an analysis on Hanoi attractors. To this end, we introduce an appropriate resistance form on the attractor, choose a suitable Radon measure and obtain a local and regular Dirichlet form that acts on the associated L2-space. This form defines a Laplacian on the Hanoi attractor, whose spectral properties we then investigate.
The study of the asymptotic behaviour of the eigenvalue counting function of this Laplacian allows us to calculate the spectral dimension of the Hanoi attractor, which turns out to coincide with the one of the Sierpinski gasket for all alpha in (0, 1/3).