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Mauro Patrão

Publications and source records attributed to Mauro Patrão.

At least 19 recordsLinked to original sources
arXiv2025

Order of dynamical and control systems on maximal compact subgroups

In this paper, we characterize the dynamical orders of minimal Morse components and, partially, of control sets on maximal compact subgroups of a semisimple Lie group as an algebraic order similar to the Bruhat order of the Weyl group. We call this order the extended Bruhat order of the extended Weyl group (or Weyl-Tits group), since it projects onto the Bruhat order and is obtained in an analogous way as the Bruhat order was historically obtained, as the incidence order of the Schubert cells in the maximal flag manifold, but now the Schubert cells in the maximal extended flag manifold.

math.DS↗
arXiv2025

The shortest way to the geodesics of spheres

In this paper, we prove, using only elementary geometric arguments and only assuming that the curves are continuous, that the geodesics on a sphere are the minor arcs of the great circles. Our result are valid for any sphere in any inner product space.

math.GM↗
arXiv2024

Dynamics of translations on maximal compact subgroups

In this article, we study the dynamics of translations of an element of a semisimple Lie group $G$ acting on its maximal compact subgroup $K$. First, we extend to our context some classical results in the context of general flag manifolds, showing that when the element is hyperbolic its dynamics is gradient and its fixed points components are given by some suitable right cosets of the centralizer of the element in $K$. Second, we consider the dynamics of a general element and characterizes its recurrent set, its minimal Morse components and their stable and unstable manifolds in terms of the Jordan decomposition of the element, and we show that each minimal Morse component is normally hyperbolic.

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arXiv2024

Cellular homology of compact groups: Split real forms

In this article, we use the Bruhat and Schubert cells to calculate the cellular homology of the maximal compact subgroup $K$ of a connected semisimple Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal compact subgroup the previously known attaching maps for the maximal flag manifold and use it to characterize algebraically the incidence order between Schubert cells. We also present algebraic formulas to compute the boundary maps which extend to the maximal compact subgroups similar formulas obtained in the case of the maximal flag manifolds. Finally, we apply our results to calculate the cellular homology of $\mbox{SO}(3)$ as the maximal compact subgroup of $\mbox{SL}(3, \mathbb{R})$ and the cellular homology of $\mbox{SO}(4)$ as the maximal compact subgroup of the split real form $G_2$.

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arXiv2024

Control sets on maximal compact subgroups

In this paper we study the action of semigroups with nonempty interior of noncompact connected semisimple Lie groups, with finite center, on their maximal compact connected subgroups. As main results we describe the set of transitivity of a control set as fixed points of regular elements in the semigroup, and determine the number of control sets.

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arXiv2024

Counting geodesics on compact symmetric spaces

We describe the inverse image of the Riemannian exponential map at a basepoint of a compact symmetric space as the disjoint union of so called focal orbits through a maximal torus. These are orbits of a subgroup of the isotropy group acting in the tangent space at the basepoint. We show how their dimensions (infinitesimal data) and connected components (topological data) are encoded in the diagram, multiplicities, Weyl group and lattice of the symmetric space. Obtaining this data is precisely what we mean by counting geodesics. This extends previous results on compact Lie groups. We apply our results to give short independent proofs of known results on compact symmetric spaces.

math.DG↗
arXiv2023

Fully geometric definitions of orientation and determinants

We present fully geometric definitions of orientation and determinants and show they coincide with the algebraic definitions. This allows us to provide an approach to determinants in the spirit of what is presented in the article A Geometric Approach to Determinants, previously published by the Monthly, but making it rigorous and complete.

math.GM↗
arXiv2021

On the embeddability of the homogeneous Ricci flow and its collapses

This article grew out of the urge to realize explicit examples of solutions for the Ricci flow as families of isometrically embedded submanifolds, together with its Gromov-Hausdorff collapses. To this aim, we consider the Ricci flow of invariant metrics in a class of flag manifolds. On the one hand, we contrast with a previous result in literature by presenting entire flow lines of invariant metrics realized as orbits of a fixed representation. Indeed, we prove that the subset of realizable metrics has a global attractor with open interior, containing an expressive family of complete flow lines. On the other hand, we prove that certain collapses cannot be realized in any fixed Euclidean space. We provide a detailed picture of the flow, including examples of both realizable and non-realizable flow lines and collapses.

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arXiv2021

The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds

The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for the homogeneous Ricci flow with natural compactness properties. In addition, we present a characterization for Gromov-Hausdorff limits of homogeneous spaces. As a result, we present a detailed picture of the homogeneous Ricci flow for three-isotropy-summands flag manifolds: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the possible Gromov-Hausdorff limits of the aforementioned lines of flow.

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arXiv2019

The Topological Entropy of Powers on Lie Groups

This article addresses the problem of computing the topological entropy of an application $ψ: G \to G$, where $G$ is a Lie group, given by some power $ψ(g) = g^k$, with $k$ a positive integer. When $G$ is commutative, $ψ$ is an endomorphism and its topological entropy is given by $h(ψ) = \dim(T(G)) \log(k)$, where $T(G)$ is the maximal torus of $G$, as shown in \cite{patrao:endomorfismos}. But when $G$ is not commutative, $ψ$ is no longer an endomorphism and these previous results cannot be used. Still, $ψ$ has some interesting symmetries, for example, it commutes with the conjugations of $G$. In this paper, the structure theory of Lie groups is used to show that $h(ψ) = \dim(T)\log(k)$, where $T$ is a maximal torus of $G$, generalizing the commutative case formula. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

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arXiv2018

The topological entropy of endomorphisms of Lie groups

In this paper, we determine the topological entropy $h(ϕ)$ of a continuous endomorphism $ϕ$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(ϕ)$ associated to a left invariant distance $d$ just provides an upper bound to $h(ϕ)$, which is characterized by the so called variational principle. We prove that \[ h\left(ϕ\right) = h\left(ϕ|_{T(G_ϕ)}\right) \] where $G_ϕ$ is the maximal connected subgroup of $G$ such that $ϕ(G_ϕ) = G_ϕ$, and $T(G_ϕ)$ is the maximal torus in the center of $G_ϕ$. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.

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arXiv2016

The minimal Morse components of translations on flag manifolds are normally hyperbolic

Consider the iteration of an invertible matrix on the projective space: are the Morse components normally hyperbolic? As far as we know, this was only stablished when the matrix is diagonalizable over the complex numbers. In this article we prove that this is true in the far more general context of an arbitrary element of a semisimple Lie group acting on its generalized flag manifolds: the so called translations on flag manifolds. This context encompasses the iteration of an invertible non-diagonazible matrix on the real or complex projective space, the classical flag manifolds of real or complex nested subspaces and also symplectic grassmanians. Without these tools from Lie theory we do not know how to solve this problem even for the projective space.

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arXiv2016

Entropy and Its Variational Principle for Locally Compact Metrizable Systems

For a given topological dynamical system $(X,T)$ over a compact set $X$ with a metric $d$, the "variational principle" states that \begin{equation*} \sup_μh_μ(T) = h(T) = h_d(T), \end{equation*} where $h_μ(T)$ is the Kolmogorov-Sinai entropy, with the supremum taken over every $T$-invariant probability measure, $h_d(T)$ is the Bowen entropy, and $h(T)$ is the topological entropy as defined by Adler, Konheim and McAndrew. In [9], the concept of topological entropy was adapted for the case where $T$ is a proper map and $X$ is locally compact separable and metrizable, and the variational principle was extended to \begin{equation*} \sup_μh_μ(T) = h(T) = \min_d h_d(T), \end{equation*} where the minimum is taken over every distance compatible with the topology of $X$. In the present work, we dropped the properness assumption, extending the above result for any continuous map $T$. We also apply our results to extend some previous formulas for the topological entropy of continuous endomorphisms of connected Lie groups proved in [4]. In particular, we prove that any linear transformation $T: V \to V$ over a finite dimensional vector space $V$ has null topological entropy.

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arXiv2014

The Isotropy Representation of a Real Flag Manifold: Split Real Forms

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the decomposition into irreducible components is not in general unique, since there are cases with infinitely many invariant subspaces.

math.DG↗
arXiv2012

Entropy of Endomorphisms of Lie Groups

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus.

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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.

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