اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDiscrete dynamics and differentiable stacks
71
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffels theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.
قيم البحث
اقرأ أيضاً
We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rig
idity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.
The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equi
valence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a $mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack ${mathfrak X}$. It turns out that shifted $(+1)$ Poisson structures on ${mathfrak X}$ correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla. (2) We introduce the notion of tangent complex $T_{mathfrak X}$ and cotangent complex $L_{mathfrak X}$ of a differentiable stack ${mathfrak X}$ in terms of any Lie groupoid $Gamma{rightrightarrows} M$ representing ${mathfrak X}$. They correspond to homotopy class of 2-term homotopy $Gamma$-modules $A[1]rightarrow TM$ and $T^vee Mrightarrow A^vee[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${mathfrak X}$, defines a morphism ${L_{{mathfrak X}}}[1]to {T_{{mathfrak X}}}$.
The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped t
o improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley index theory for multivalued maps [B. Batko and M. Mrozek. Weak index pairs and the Conley index for discrete multivalued dynamical systems, SIAM J. Applied Dynamical Systems 15 (2016), 1143-1162], [B.Batko. Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the Index, SIAM J. Applied Dynamical Systems 16 (2017), 1587-1617] is used in computations. The existence of a continuous, single-valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of Henon type maps are presented.
In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the
associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck-Witt group of a derived stack, and compare it to the classical one.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
