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Victor Ayala

Publications and source records attributed to Victor Ayala.

29 records · Page 2Linked to original sources
arXiv2018

Dynamics on Hyperspaces

Given a compact metric space (X; \varrho) and a continuous function f:X\rightarrow X, we study the dynamics of the induced map \bar{f} on the hyperspace of the compact subsets of X. We show how the chain recurrent set of f and its components are related with the one of the induced map. The main result of the paper proves that, under mild conditions, the numbers of chain components of \bar{f} is greater than the ones of f. Showing the richness in the dynamics of \bar{f} which cannot be perceived by f.

math.DS↗
arXiv2018

Affine and bilinear systems on Lie groups

In this paper we study affine and bilinear systems on Lie groups. We show that there is an intrinsic connection between the solutions of both systems. Such relation allows us to obtain some preliminary controllability results of affne systems on compact and solvable Lie groups. We also show that the controllability property of bilinear systems is very restricted and may only be achieved if the state space G is an Euclidean space.

math.DS↗
arXiv2017

Topological entropy of a Lie group automorphism

Let ϕ be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of ϕ we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a ϕ invariant Levi subgroup. Then we prove that the topological entropy information of ϕ is contained in the toral component of the unstable subgroup of ϕ in the radical of G. We specialize the main result in a couple of interesting situations.

math.DS↗
arXiv2017

Isometries of almost-Riemannian structures on Lie groups

A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.

math.OC↗
arXiv2016

A Semigroup associated to a linear control system on a Lie group

Let us consider a linear control system Σon a connected Lie group G. It is known that the accessibility set A from the identity e is in general not a semigroup. In this article we associate a new algebraic object S to Σwhich turns out to be a semigroup, allowing the use of the semigroup machinery to approach Σ. In particular, we obtain some controllability results.

math.DS↗
arXiv2016

Control Systems on Flag Manifolds and their chain Control sets

In this paper we shown that the chain control sets of induced systems on flag manifolds coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set.

math.DS↗
arXiv2016

Control sets of linear systems on Lie groups

Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies the properties of the maximal sets of approximate controllability.

math.DS↗
arXiv2015

Almost-Riemannian Geometry on Lie Groups

A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARS's, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.These ARS's are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous space is stated.

math.DG↗
arXiv2013

Topological Conjugacy of Real Projective Flows

In this paper we prove the following topological classification result for flows on real projective space induced by linear flows on Euclidean space: Two flows on the projective space P(V) of a finite-dimensional real vector space V, induced by endomorphisms A and B of V, are topologically conjugate if and only if the Jordan structures of A and B coincide except for the real parts of the eigenvalues whose values may differ but whose order and multiplicities must agree. Our proof is mainly based on ideas of Kuiper who considered the discrete-time analogue of this classification problem. We also correct a mistake in Kuiper's proof.

math.DS↗