Searcharxiv⌕ Search

arXiv subjects

Luiz A. B. San Martin

Publications and source records attributed to Luiz A. B. San Martin.

24 records · Page 2Linked to original sources
arXiv2014

Conditions for Equality between Lyapunov and Morse Decompositions

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow $ϕ$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$. A continuous one given by the finest Morse decomposition and a measurable one furnished by the Multiplicative Ergodic Theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under pertubations leaving unchanged the flow on the base space.

math.DS↗
arXiv2014

Differentiability of Lyaupnov Exponents

We prove differentiability of certain linear combinations of the Lyapunov spectra of a flow on a principal bundle of a semi-simple Lie group. The specific linear combinations that yield differentiability are determined by the finest Morse decomposition on the flag bundles. Differentiability is taken with respect to a differentiable structure on the gauge group, which is a Banach-Lie group.

math.DS↗
arXiv2011

Orientability of vector bundles over real flag manifolds

We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided.

math.DG↗
arXiv2011

Controllability of control systems simple Lie groups and the topology of flag manifolds

Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group $G$. It is proved that $S=G$ if $S$ contains a subgroup $G(α) \approx \mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm α}$, where $\mathfrak{g}%_α$ is the root space of the root $α$. The proof uses the fact, proved before, that the invariant control set of $S$ is contractible in some flag manifold if $S$ is proper, and exploits the fact that several orbits of $G(α)$ are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.

math.OC↗
arXiv2007

Morse and Lyapunov Spectra and Dynamics on Flag Bundles

This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group $G$. Projection against the Iwasawa decomposition $G = KAN$ defines an additive cocycle over the flow with values in $\frak{a} = \log A$. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

math.DS↗