Let $\sigma$ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_1,...,X_d$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph$𝒞$ of $\sigma$. If $\sigma$ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_1,...,\partial X_d$. From this graph...

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.