Ricardo Severino

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.

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 sequencern 𝒦(fμ) of the one-parameter familyrnof real quadratic maps fμ.

Polynomials over Quaternions and Coquaternions: A Unified Approach

This paper aims to present, in a unified manner, results which are valid on both the algebras of quaternions and coquaternions and, simultaneously, call the attention to the main differences between these two algebras. The rings of one-sided polynomials over each of these algebras are studied and some important differences in what concerns the structure of the set of their zeros are remarked. Examples illustrating this different behavior of the zero-sets of quaternionic and coquaternionic polynomials are also presented.

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 .

Weierstrass method for quaternionic polynomial root-finding

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass-like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

Quaternionic polynomials with multiple zeros: A numerical point of view

COMPETE reference number POCI-01-0145-FEDER-006683; FCT/MECs financial support through national funding and by the ERDF through the Operational Programme on ``Competitiveness and Internationalization -- COMPETE 2020 under the PT2020 Partnership Agreement

THE BASIN OF ATTRACTION OF LOZI MAPPINGS

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

COMPUTING TOPOLOGICAL INVARIANTS IN BOUNDARY VALUE PROBLEMS REDUCIBLE TO DIFFERENCE EQUATIONS

Among boundary values problems (BVP) for partial differential equations there are certain classes of problems reducible to difference equations. Effective study of such problems has became possible in the last 20-30 years owing to appreciable advances done also in the theory of difference equations with discrete time, specifically given by one-dimensional maps. Here we apply how this reduction method may be used in simple nonlinear BVP, determined by a bimodal map. We consider two-dimensional linear hyperbolic system with constant coefficients, with nonlinear boundary conditions and usual initial conditions. The objective is to characterize the dependence of the motions of the vortice solutions with the topological invariants of the bimodal map.

The number of zeros of unilateral polynomials over coquaternions revisited

Abstract The literature on quaternionic polynomials and, in particular, on methods for finding and classifying their zero sets, is fast developing and reveals a growing interest in this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovská and Opfer [Electron Trans Numer Anal. 2017;46:55–70], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, zeros. In this paper we present a full proof of this result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.