J. Sousa Ramos

Isentropic Real Cubic Maps

Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. With this work, we want to understand how to use this second invariant to distinguish bimodal maps with the same topological entropy and, in particular, how this second invariant changes within a given type of topological entropy level set. In order to do that, we use the kneading theory framework and introduce a symbolic product * between kneading invariants of maps from the same topological entropy level set, for which we show that the second invariant is preserved. Finally, we also show that the change of the second invariant follows closely the symbolic order between bimodal kneading sequences.

Topological invariants and renormalization of Lorenz maps

We prove that the invariants of the topological semiconjugation of Lorenz maps with β-transformations remains constant on the renormalization archipelagoes and analyze how the dynamics on the archipelagoes depends on its structure.

Kneading theory for tree maps

Using the techniques developed in [ASR], we generalize to tree maps the Milnor and Thurston results about zeta function, semiconjugacy and topological entropy of interval maps. (Received 20 Fev. 2001)

WEIGHTED KNEADING THEORY OF ONE-DIMENSIONAL MAPS WITH A HOLE

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

K-theory for Cuntz-Krieger algebras arising from real quadratic maps

We compute the K-groups for the Cuntz-Krieger algebras 𝒪A𝒦(fμ), where A𝒦(fμ) is the Markov transition matrix arising from the kneading sequence 𝒦(fμ) of the one-parameter family of real quadratic maps fμ.

One-Dimensional Semiconjugacy Revisited

Let f be a piecewise monotone interval map with positive topological entropy h(f)=log(s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

The order of chaos on a Bianchi-IX cosmological model

The purpose of this paper is to analyze the chaotic behavior that can arise on a type-IX cosmological model using methods from dynamic systems theory and symbolic dynamics. Specifically, instead of the Belinski-Khalatnikov-Lifschitz model, we use the iterates of a monotonously increasing map of the circle with a discontinuity, and for the Hamiltonean dynamics of Misners Mixmaster model we introduce the iterates of a noninvertible map. An equivalence between these two models can easily be brought upon by translating them in symbolic-dynamical terms. The resulting symbolic orbits can be inserted in an ordered tree structure set, and so we can present an effective counting and referentation of all period orbits.

Irreducible complexity of iterated symmetric bimodal maps

We introduce a tree structure for the iterates of symmetric bimodal maps and identify a subset which we prove to be isomorphic to the family of unimodal maps. This subset is used as a second factor for a ∗ -product that we define in the space of bimodal kneading sequences. Finally, we give some properties for this product and study the ∗ -product induced on the associated Markov shifts.

Bowen-Franks Groups for Bimodal Matrices

From the symbolic description of iterated bimodal maps of the interval, we can associate such maps to square 0, 1 transition matrices, referred as bimodal matrices. We show that the Bowen-Franks group BF ( A ) of a bimodal matrix A is given Z c ] Z d , for some non-negative integers c , d .

Boundary Maps and Fenchel–Nielsen–Maskit Coordinates

We consider a genus 2 surface, M, of constant negative curvature and we construct a 12-sided fundamental domain, where the sides are segments of the lifts of closed geodesics on M (which determines the Fenchel–Nielsen–Maskit coordinates). Then we study the linear fractional transformations of the side pairing of the fundamental domain. This construction gives rise to 24 distinct points on the boundary of the hyperbolic covering space. Their itineraries determine Markov partitions that we use to study the dependence of the Lyapunov exponent and length spectrum of the closed geodesics with the Fenchel–Nielsen coordinates.