Víctor F. Sirvent

On 2-Reptiles in the Plane

We classify all rational 2-reptiles in the plane. We also establish properties concerning rational reptiles in the plane in general.

Identifications and Dimension of the Rauzy Fractal

In this paper we describe explicitly the identifications on the boundary of the Rauzy fractal that makes it a fundamental domain of the two-dimensional torus, for the action of the lattice Z2 on the plane. Using these identifications, we give a new way to compute the Hausdorff dimension of the Rauzy fractal. Also, we compute the Hausdorff dimension of the realization of the boundary of the Rauzy fractal on the interval.

Geodesic laminations as geometric realizations of Pisot substitutions

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution:

On some dynamical subsets of the Rauszy fractal

1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1

Relationships between the dynamical systems associated to the Rauzy substitutions

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

Let (Nn,(+1)n) be the adic system associated to the substitution: 1 → 12,…,(n − 1) → 1n, n → 1. In Sirvent (1996) it was shown that there exist a subset Cn of Nn and a map hn: C → Cn such that the dynamical system (C, hn) is semiconjugate to (Nn−1,(+1)n−1). In this paper we compute the Hausdorff and Billingsley dimensions of the geometrical realizations of the set Cn on the (n− l)-dimensional torus. We also show that the dynamical system (Cn,hn) cannot be realized on the (n − 1)-torus.

Cyclotomic polynomials and minimal sets of Lefschetz periods

Abstract In this paper we study some relationships between the dynamical systems that arise from the family of substitutions: 1 → 12, …,( n − 1) → 1 n , n → 1, for n ⩾ 2. We describe how the dynamics of the system of this family corresponding to n is present in the system corresponding to n ′, with n n ′.

Automatic β-expansions of formal Laurent series over finite fields

We study the minimal set of (Lefschetz) periods of the C 1 Morse–Smale 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 kinds of results. First, we completely characterize the possible minimal sets of periods for the C 1 Morse–Smale diffeomorphisms on non-orientable compact surface without boundary of genus g with . But the proof of these results provides an algorithm for characterizing the possible 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.

Dynamical properties of the tent map

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the s are positive integers, s are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855).

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus

We consider @b-expansions of formal Laurent series over finite fields. If the base @b is a Pisot or Salem series, we prove that the @b-expansion of a Laurent series @a is automatic if and only if @a is algebraic.