| 词汇 | example_english_manifold |
| 释义 | Examples of manifoldThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. Cycles for the dynamical study of foliated manifolds and complex manifolds. Families of invariant manifolds corresponding to non-zero characteristics exponents. Here objects are compact oriented manifolds (boundaries), and morphisms are oriented bordisms (space-times). All known manifolds with engaging actions of higher rank lattices have an arithmetic fundamental group. We want also to associate manifolds with boundary to graphs with boundary. One often considers oriented bordisms, which are oriented manifolds with boundary. The statement of (2.9) only relies on the topological concepts of orientation and boundaries of manifolds. Similar definitions hold for functions taking values in manifolds. Conjugacy and rigidity for nonpositively curved manifolds of higher rank. Thus, is supported on a set which contains many full unstable manifolds. We prove that such stable manifolds have their sizes bounded from below for infinitely hyperbolic points. The following proposition describes the local stable and unstable manifolds. We show that the metric of non-positively curved graph manifolds is determined by its geodesic flow. Since any quasi-isometry between negatively curved simply connected manifolds induces a quasiconformal map of the boundaries, we have the following. We study the growth of unstable manifolds of automorphisms of complex surfaces. The motivation of the previous theorem is the classical problem of denseness of smooth conservative diffeomorphisms of compact manifolds. Let us recall first the definition of geometric structures on manifolds. Such manifolds (if they exist) would provide examples where (6.6) yields a better estimate than (6.10). In the process we saw that there were examples of non-isometric graph manifolds of non-positive curvature whose universal covers have equivalent boundaries. We consider dynamical systems on compact manifolds that are local diffeomorphisms outside an exceptional set (a compact submanifold). The stable and unstable manifolds of the saddletype motions may intersect transversely, yielding chaotic fluid particle motions. Quasiconformality in the geodesic flow of negatively curved manifolds. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds [9]. Families of invariant manifolds corresponding to nonzero characteristic exponents. A detailed study of local stable and unstable manifolds was also carried out in [108]. The fact is that the topological sets we are studying, that is, the unstable manifolds of one-rectangle systems, have a dynamical origin. The reference set is the union of a finite number of 'rectangles' whose sides are parallel to unstable and stable manifolds. We will refer to these as the integral manifolds. In particular, we take advantage of the theory of intersections of complex manifolds to analyze the complex extensions of the real stable and unstable manifolds. Numerical algorithms are incorporated in the class of flows on manifolds. The sets of points approaching an invariant set as t + sometimes are called stable and unstable manifolds, respectively. We then devise a procedure for lifting these curves, that efficiently cover the unit cube, to abstract surfaces, such as nonlinear manifolds. The desired dynamics is achieved by the selection of these manifolds. From these computations we will show that at the critical values the integral manifolds undergo bifurcations in their topology. A property of compact, connected, laminated subsets of manifolds. The marked point lies on two manifolds and thus has two degenerate subconfigurations. A system in which one of these three conditions did not hold for all invariant manifolds could not be conformal symplectic. The stable and unstable manifolds of a hyperbolic set are injectively immersed submanifolds. The picture indicates the manifolds for a return map on a global cross-section. The center manifolds are transverse intersections of center stable manifolds with center unstable manifolds. The invariant curves are obtained as intersections of center stable with center unstable manifolds. However, with the exception of these maps, the dynamics of arbitrary continuous maps of manifolds have not been extensively studied. We generalize the concept of horospherical points, first defined for geodesic flows on negatively curved manifolds, to flows admitting a local product structure. The arguments used above have implications for the locations of the stable and unstable manifolds of the fixed points. Thus, by manifolds we mean both manifolds without boundary and manifolds with boundary. The second numerical method is for plotting one-dimensional stable and onedimensional unstable manifolds. One concludes by the continuity and the compactness of the local strong unstable manifolds. Our main objective is to find the weakest possible hypotheses under which one can construct center manifolds. Both theorems are proved by establishing a dichotomy for the conditional measures of along the intersection of suitable stable manifolds. Upon perturbation the manifolds become separated as seen from the experiments. Using these invariant manifolds, we can give the following definitions. In general, such a result does not hold for non-compact manifolds. In what follows we will define notions of transverse dimensions along unstable manifolds. We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Here we are considering non-invariant admissible unstable manifolds, but the principles of the proof are the same as for invariant manifolds. The simplest of such manifolds is the sphere with the standard metric. The structure of flows exhibiting nontrivial recurrent on two-dimensional manifolds. A cycle consists only of heteroclinic and fixed points which are corner points on an alternating sequence of branches of stable and unstable manifolds. Closing stable and unstable manifolds in the two-sphere. We know of no such results for homeomorphisms of non-compact manifolds. Part (i) was proved in [17] (for manifolds with finitely many ends) and more generally in [19]. There are a number of results on compact manifolds with boundary or on finite-volume manifolds. A version of the smooth realization problem addresses the possibility of obtaining certain properties invariant under measurable conjugacy for volume-preserving diffeomorphisms of compact manifolds. Roughly speaking, they obtain topological results about certain manifolds provided they admit special codimension 1 immersions. The present paper develops a new calculus with global projection data for operators on manifolds with edges. Here are some elementary observations about manifolds of finite type. Thus, 2-dimensional proof-structures are manifolds associated to 1-dimensional proof-structures in the sense of our 'new' definition. A strange attractor belongs to a class of attractors that do not lie on manifolds. Manifolds without focal points are examples of such manifolds. Two objects are crucial in this description: invariant manifolds and, in the dissipative case, attracting sets and, in particular, strange attractors. The goal of this section is to introduce a new type of transversality between invariant manifolds of a fixed point. Geodesic flows on manifolds of negative constant curvature. When the manifolds are negatively curved closed surfaces, it is shown in [20] that the marked length spectrum determines the geometry of the surface. We next consider if the planar manifolds can serve as boundaries for cross sections to the spatial manifolds. Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. We consider partially hyperbolic diffeomorphisms on compact manifolds. They consist of parallel line segments, orthogonal to the stable manifolds, and their length is measurable. All the stable manifolds will therefore be parallel line segments but the length of the manifolds are only measurable [12]. A rigidity theorem for simply connected manifolds without conjugate points. Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. The question of twist maps was originally investigated in manifolds of dimension two (see for example [6, 7]). The diffeomorphism f satisfies the transversality condition if the stable and unstable manifolds of the nonwandering set intersect transversally. One can also apply these results to geodesic flows on manifolds of constant negative curvature. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate. The third is 'precariousness', or its position relative to the thresholds - the unstable manifolds between stability domains. Along each of the strong unstable manifolds there exists what can be thought of as an 'infinitesimal foliation', or invariant 'transverse derivatives'. To finish the proof just note that the projections of any two different two-dimensional unstable manifolds cannot intersect. In this case, the situation is much more complicated: one-dimensional unstable manifolds project into two circles, each one corresponding to a separatrix. We only need that the return maps at periodic orbits are diagonal on the stable and unstable manifolds. The latter one has the advantage that when the stable or unstable manifolds are one-dimensional, it makes step unnecessary. Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities. However, we can choose a continuous family of locally invariant manifolds tangent to it. The proof is inspired by the arguments in, although that paper considers only stable and unstable invariant manifolds. However, to the best of our knowledge, there exist no results in the literature considering the case of center manifolds in the non-uniform setting. Furthermore, this method works equally well for the dimension of a hyperbolic set on unstable manifolds. From this we derive applications to geodesic flows on manifolds of constant negative curvature. We know that has measure one, and that has absolutely continuous conditional measures on unstable manifolds. Manifolds with no focal points satisfy this axiom, and in particular manifolds of non-positive curvature. To overcome this we will consider only local stable manifolds and proceed inductively. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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