51,223 research outputs found
Completely reducible sets
International audienceWe study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets
Completely reducible hypersurfaces in a pencil
We study completely reducible fibers of pencils of hypersurfaces on and associated codimension one foliations of .
Using methods from theory of foliations we obtain certain upper bounds for
the number of these fibers as functions only of .
Equivalently this gives upper bounds for the dimensions of resonance
varieties of hyperplane arrangements.
We obtain similar bounds for the dimensions of the characteristic varieties
of the arrangement complements.Comment: 15 pages, 2 figure
Period doubling and reducibility in the quasi-periodically forced logistic map
We study the dynamics of the Forced Logistic Map in the cylinder. We compute
a bifurcation diagram in terms of the dynamics of the attracting set. Different
properties of the attracting set are considered, as the Lyapunov exponent and,
in the case of having a periodic invariant curve, its period and its
reducibility. This reveals that the parameter values for which the invariant
curve doubles its period are contained in regions of the parameter space where
the invariant curve is reducible. Then we present two additional studies to
explain this fact. In first place we consider the images and the preimages of
the critical set (the set where the derivative of the map w.r.t the
non-periodic coordinate is equal to zero). Studying these sets we construct
constrains in the parameter space for the reducibility of the invariant curve.
In second place we consider the reducibility loss of the invariant curve as
codimension one bifurcation and we study its interaction with the period
doubling bifurcation. This reveals that, if the reducibility loss and the
period doubling bifurcation curves meet, they do it in a tangent way
Cocharacter-closure and spherical buildings
Let be a field, let be a reductive -group and an affine
-variety on which acts. In this note we continue our study of the notion
of cocharacter-closed -orbits in . In earlier work we used a
rationality condition on the point stabilizer of a -orbit to prove Galois
ascent/descent and Levi ascent/descent results concerning cocharacter-closure
for the corresponding -orbit in . In the present paper we employ
building-theoretic techniques to derive analogous results.Comment: 16 pages; v 2 17 pages, exposition improved; to appear in the Robert
Steinberg Memorial Issue of the Pacific Journal of Mathematic
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