Properties of geometrical realizations of substitutions associated to a family of Pisot numbers
In this thesis we study some properties of the geometrical realizations of the dynalnical systelns that arise from the family of Pisot substitutions: 1 →- 12 11n : 2 → 13 (n-1) → 1n n →1 for n a positive integer greater than 2. In chapter 1 we COlllpute the Holder exponent of the Arnoux Inap, which is the selniconjugacy between the geollletrical realization of (n, 0'), the dynaIllical systelll of this substitution, in the circle (SI, f) and the 11, - 1 dilnensional torus (Tn-I, T). Also in this chapter we introduce the notion of the standard partition in the SYlllbolic space n and in its geollletrical realizations. The cylinders of this partition are classified according to their structure. In chapter 2 we construct a geodesic lamination on the hyperbolic disk associated to this standard partition and a transverse Ineasure on the laInination. The interval exchange Inap f and the contraction h induce Inaps F and H on the lan,lination, respectively. The map .F preserves the transverse Ineasure and H contracts it. In chapter 3 we cOInpute the Hausdorff dimension of the boundary of w,. the fundalnental dOlnain of the torus T2 obtained by the realization of the sYInbolic space n that arises froln the substitution 113 • As a corollary we cOInpute the Hausdorff dhnension of the pre-image· of the bo:undary of w under the Arnoux Inap. 'Ve also describe the identifications on the boundary of w that Inake it a fundalllental dOlnain of the hvo dilnensional torus. In chapter 4 we study some relationships between the dynamical systeIlls of this faIllily of sub,stitutions. "Te describe how the dynamics of the systeIlls of this faIllily, corresponding to lower dimensions - i.e. the parallleter n in the definition of lIn - are present in systems of higher dimensions. Also we study the realization of this property in the interval'.