weakly chaotic functions with zero topological entropy and non- flat critical points.
-algebras associated with presentations of subshifts.
Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey
Cardinality of Rauzy classes
Rauzy classes form a partition of the set of irreducible permutations. They were introduced as part of a renormalization algorithm for interval exchange transformations. We prove an explicit formula for the cardinality of each Rauzy class. Our proof uses a geometric interpretation of permutations and Rauzy classes in terms of translation surfaces and moduli spaces.
Chaos & pseudochaos: Some basic remarks
Chaos and geometric order in architecture and design.
Chaos and shadowing around a homoclinic tube.
Chaos in nonautonomous dynamical systems.
Chaos made visual.
In this paper we show how the main properties of chaos can be fully visualized at the light of a very easy to handle object, the tent function. Although very concrete, this case is representative of a very large number of examples, with more or less the same properties.
Chaotic behaviour of the map x ↦ ω(x, f)
Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and...
Characterization of chaos for continuous maps of the circle
Characterization of substitution invariant words coding exchange of three intervals.
Characterization of -limit sets of continuous maps of the circle
In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of -limit sets from the class of continuous maps of the interval to the class of continuous maps of the circle. Among others we give geometric characterization of -limit sets and then we prove that the family of -limit sets is closed with respect to the Hausdorff metric.
Choosing an attacker by a local derivation
Classification of recurrent domains for some holomorphic maps.
Codages de rotations et phénomènes d'autosimilarité
Nous étudions une classe de suites symboliques, les codages de rotations, intervenant dans des problèmes de répartition des suites et représentant une généralisation géométrique des suites sturmiennes. Nous montrons que ces suites peuvent être obtenues par itération de quatre substitutions définies sur un alphabet à trois lettres, puis en appliquant un morphisme de projection. L’ordre d’itération de ces applications est gouverné par un développement bi-dimensionnel de type “fraction continue”...
Coexisting cycles in a class of 3-D discrete maps
In this paper we consider the class of three-dimensional discrete maps M (x, y, z) = [φ(y), φ(z), φ(x)], where φ : ℝ → ℝ is an endomorphism. We show that all the cycles of the 3-D map M can be obtained by those of φ(x), as well as their local bifurcations. In particular we obtain that any local bifurcation is of co-dimension 3, that is three eigenvalues cross simultaneously the unit circle. As the map M exhibits coexistence...
Coincidence for substitutions of Pisot type
Let be a substitution of Pisot type on the alphabet ; satisfies thestrong coincidence conditionif for every , there are integers such that and have the same -th letter, and the prefixes of length of and have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if and provide a partial result for .
Collective geodesic flows
We show that most compact semi-simple Lie groups carry many left invariant metrics with positive topological entropy. We also show that many homogeneous spaces admit collective Riemannian metrics arbitrarily close to the bi-invariant metric and whose geodesic flow has positive topological entropy. Other properties of collective geodesic flows are also discussed.
Combinatorial mapping-torus, branched surfaces and free group automorphisms
We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a -cocycle of a -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...