We classify a quasiperiodic flow as either algebraic or transcendental. For an algebraic quasiperiodic flow on the n-torus, we prove that an absolute invariant of the smooth conjugacy class of this flow, known as the multiplier group, is a subgroup of the group of units of the ring of integers in a real algebraic number field of degree n over Q. We also prove that for any real algebraic number field F of degree n over Q, there exists an algebraic quasiperiodic flow on the n-torus whose multiplier group is exactly the group of units of the ring of integers in F. We support and formulate the conjecture that the multiplier group distinguishes the algebraic quasiperiodic flows from the transcendental ones.
Our interest in this work is in group extensions of minimal flows with compact abelian groups in the fibres. We study their structure from categorical and algebraic points of view, and describe relations of their dynamics to the one-dimensional algebraic-topological invariants. We determine the first cohomology groups of flows with simply connected acting groups and those of topologically free flows possessing a free cycle. As an application we show that minimal extensions of these flows not only do exist, but they have a rich algebraic structure.
We study flows of smooth vector fields $X$ over invariant surfaces $M$ which are levels of rational first integrals. It leads us to study constrained systems, that is, systems with impasses. We identify a subset $\mathcal{I} \subset M$ which we call "pseudo-impasse" set and analyze the flow of X by points of $\mathcal{I}$. Systems well known in the literature exemplify our results: Lorenz, Chen, Falkner-Skan and Fisher-Kolmogorov. We also study 1-parameter families of integrable systems and unfolding of minimal sets. Our main tool is the geometric singular perturbation theory.
The linear flows in the space of Krichever-Lax matrices over an algebraic curve
在\cite{kri02}中，I. M. Krichever在紧Riemann曲面上建立了参数化稳定向量丛的模空间的余切丛的矩阵空间，并在研究\cite{hit87}后将其命名为Hitchin系统。本文利用P.A.的工作\cite{grif85}中类似的技巧和分析，研究了Lax表示中Krichever-Lax矩阵空间上的流在上同调类意义下线性的一个充要条件。格里菲斯
In this article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation $(X,v)$, based on the study of rational factors of $(X,v)$ and of algebraic foliations on $X$, invariant under the Lie-derivative of the vector field $v$. We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disint…
Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $Γ$ be a lattice in $G$, with $π:G\to G/Γ$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $π(X)$ in the compact nilmanifold $G/Γ$. Our theorem describes $\mathrm{cl}(π(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $Γ$. We al…
