Symmetric intersections of Rauzy fractals

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions

C1 self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Altres ajuts: ICREA AcademiaLet X be a connected closed manifold and f a self-map on X. We say that f is almost quasi-unipotent if every eigenvalue λ of the map f∗k (the induced map on the k-th homology group of X) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of λ as eigenvalue of all the maps f∗k with k odd is equal to the sumof the multiplicities of λ as eigenvalue of all the maps f∗k with k even. We prove that if f is C1 having finitely many periodic points all of them hyperbolic, then f is almost quasi-unipotent

Periods of holomorphic maps on compact Riemann surfaces and product of spheres

In this article, we consider non-constant holomorphic maps on Riemann surfaces and product of Riemann spheres, and we give conditions on the maps in order that they have arbitrary large prime numbers as periods. We use Lefschetz fixed point theory and in particular we compute the Lefschetz numbers of period m for large m's

Periodic structure of transversal maps on sum-free products of spheres

In this article we study the periodic structure of transversal maps on the product of spheres of different dimensions. In particular we give conditions for the maps to have infinitely many even and odd periods. Moreover we give conditions for having non-zero Lefschetz numbers of period m, for infinitely many m's. We generalize the results for transversal maps on rational exterior spaces of rank 1

Periodic structure of the transversal maps on surfaces

In this article we study the set of periods of transversal maps on orientable and non-orientable compact surfaces without boundary. We provide sufficient conditions, in terms of the spectra of the induced maps on homology, in order that the map has infinitely many periods, in particular odd periods

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 dynamical systems that arise from the family of Pisot substitutions: 1 → 12 IIn :2 → 13 : (n −1) → 1n n → 1 for n a positive integer greater than 2. In chapter 1 we compute the Holder exponent of the Arnoux map, which is the semiconjugacy between the geometrical realization of (Ω ό), the dynamical system of this substitution, in the circle (SI, f) and the n – 1 dimensional torus (Tn-I, T). Also in this chapter we introduce the notion of the standard partition in the symbolic space Ω and in its geometrical 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 measure on the lamination. The interval exchange map f and the contraction h induce maps F and H on the lamination, respectively. The map .F preserves the transverse measure and H contracts it. In chapter 3 we compute the Hausdorff dimension of the boundary of w,. the fundamental domain of the torus T2 obtained by the realization of the symbolic Ω space that arises from the substitution II3. As a corollary we compute the Hausdorff dimension of the pre-image·of the boundary of w under the Arnoux map. We also describe the identifications on the boundary of w that make it a fundamental domain of the two dimensional torus. In chapter 4 we study some relationships between the dynamical systems of this family of substitutions. We describe how the dynamics of the systems of this family corresponding to lower dimensions - i.e. the parameter n in the definition of IIn - present in systems of higher dimensions. Also we study the realization of this property in the interval

On Lefschetz periodic point free self-maps

We study the periodic point free maps on connected retract of a finite simplicial complex using the Lefschetz numbers. We put special emphasis in the self-maps on the product of spheres and of the wedge sums of spheres

Minimal sets of periods for Morse-Smale diffeomorphisms on non-orientable compact surfaces without boundary

We study the minimal set of (Lefschetz) periods of the C1 Morse-Smale diffeomorphisms on a non-orientable compact surface without boundary inside its class of homology. In fact our study extends to the C1 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 C1 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 C1 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 C1 Morse-Smale diffeomorphisms on a non-orientable compact surface without boundary