I will present some of the results by Marcy Barge and Jaroslaw Kwapisz (based on their paper "Geometric Theory of unimidular Pisot substitutions", Amer. J. Math., vol. 128 (2006), no. 5, pp. 1219--1282).

There are two classical ways of studying substitution tilings of the line: symbolic dynamics, and endomorphisms of ``train tracks". The authors give a strikingly new geometric approach and in particular show that if the tiling has a unimodular Pisot matrix of dimension d, then there is a factor onto the d-dimensional torus. In fact, they have a preprint removing the unimodular assumption. I propose to begin defining tilings and the tiling space X of a tiling T. X is a compact metric space that contains all tilings which have the same local patterns as T. In dimension 1 (the subject of this talk) X is similar to a solenoid.

I will not assume any familiarity with tiling theory.

In 1969, Hirsch posed the following problem: given a diffeomorphism of a manifold and a hyperbolic set for the diffeomorphism, describe the topology of the hyperbolic set and the dynamics of the diffeomorphism for this set. We solve the problem when the hyperbolic set is a closed 3-manifold.