Related papers: Jordan decomposition and dynamics on flag manifold…

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the…

In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials…

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,…

Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and…

In this paper, we generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of regular index filtration for every Morse decomposition.

A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data-driven dynamics. In the present paper we continue the program by studying…

In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems.…

The Conley index theory is a powerful topological tool for describing the basic structure of dynamical systems. One important feature of this theory is the attractor-repeller decomposition of isolated invariant sets. In this decomposition,…

Let $X$ be an open subset of $\Bbb C^N$, and let $A$ be an $n\times n$ matrix of holomorphic functions on $X$. We call a point $\xi\in X$ $\mathbf{Jordan}$ $\mathbf{stable}$ for $A$ if $\xi$ is not a splitting point of the eigenvalues of…

We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r…

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain…

Let G be a totally disconnected, locally compact group admitting a contractive automorphism f. We prove a Jordan-Holder theorem for series of f-stable closed subgroups of G, classify all possible composition factors and deduce consequences…

Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare…

We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several…

For a given ring $\mathfrak{R}$ and a locally finite pre-ordered set $(X, \leq)$, consider $I(X, \mathfrak{R})$ to be the incidence algebra of $X$ over $\mathfrak{R}$. Motivated by a Xiao's result which states that every Jordan derivation…

Modal decompositions such as proper orthogonal decomposition (POD), dynamic mode decomposition (DMD) and their variants are regularly used to educe physical mechanisms of nonlinear flow phenomena that cannot be easily understood through…

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan…

We study the rational homology of the Deligne--Mumford compactification $\overline{\mathcal M}_{g,n}$ of the moduli space of stable curves via a family of Morse functions, namely the $\text{sys}_T$ functions. Exploiting the geometric and…

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this…

Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this…