In this paper we show that a class of sets known as the Rauzy fractals, which are constructed via substitution dynamical systems, give rise to self-affine multi-tiles and self-affine tilings. This provides an efficient and unconventional way for constructing aperiodic self-affine tilings. Our result also leads to a proof that a Rauzy fractal R associated with a primitive and unimodular Pisot substitution has nonempty interior.

Abstract. We consider two dynamical systems associated with a substitution of Pisot type: the usual Zaction on a sequence space, and the R-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the "balanced pairs algorithm" and the "overlap algorithm") and study the relation between them. In particular, we show that pure discrete spectrum for the R-action implies pure discrete spectrum for the Z-action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every R-action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

In this paper we study a class of connected fractals that admit a space filling curve. We prove that these curves are Hölder continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.

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Abstract. We study the minimal set of (Lefschetz) periods of the C 1 MorseSmale diffeomorphisms on a non-orientable compact surface without boundary inside its class of homology. In fact our study extends to the C 1 diffeomorphisms on these surfaces having finitely many periodic orbits all of them hyperbolic and with the same action on the homology as the Morse-Smale diffeomorphisms.We mainly have two kind of results. First we completely characterize the minimal sets of periods for the C 1 Morse-Smale diffeomorphisms on non-orientable compact surface without boundary of genus g with 1 ≤ g ≤ 9. But the proof of these results provides an algorithm for characterizing these minimal sets of periods for the C 1 Morse-Smale diffeomorphisms on non-orientable compact surfaces without boundary of arbitrary genus.Second we study what kind of subsets of positive integers can be minimal sets of periods of the C 1 Morse-Smale diffeomorphisms on a non-orientable compact surface without boundary.