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Benito Pires

Publications and source records attributed to Benito Pires.

22 records · Page 2Linked to original sources
arXiv2011

Orbit structure of interval exchange transformations with flip

A sharp bound on the number of invariant components of an interval exchange transformation is provided. More precisely, it is proved that the number of periodic components n_per and the number of minimal components n_min of an interval exchange transformation of n intervals satisfy n_per+2 n_min\le n. Besides, it is shown that almost all interval exchange transformations are typical, that is, have all the periodic components stable and all the minimal components robust (i.e. persistent under almost all small perturbations). Finally, we find all the possible values for the integer vector (n_per, n_min) for all typical interval exchange transformation of n intervals.

math.DS↗
arXiv2011

Vector fields whose linearisation is Hurwitz almost everywhere

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided: Let $X:R^2-->R^2$ be a C^1 vector field whose derivative DX(p) is Hurwitz for almost all p in $R^2$. Then the singularity set of X, Sing(X), is either an emptyset, a one--point set or a non-discrete set. Moreover, if Sing(X) contains a hyperbolic singularity then X is topologically equivalent to the radial vector field $(x,y)--> (-x,-y)$. This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.

math.DS↗
arXiv2006

On Peixoto's Conjecture for Flows on Non-orientable 2-Manifolds

Let M be a non-orientable compact 2-manifold of genus 4. Then there exists a family of quasi-minimal, Kupka-Smale smooth vector fields X_r in M, depending smoothly on 0<=r<e, such that, for some flow box V in M of X_0, and for all 0<=r,v<e, (1) X_r restricted to V is a flow box; (2) If r is not equal to v, X_r is a C^\infty--twist perturbation of X_v localized in V; (3) X_r and X_v are topologically equivalent.

math.DS↗