One-parameter Family Of Diffeomorphisms Research Articles

Regularity estimates for the gradient flow of a spinorial energy functional

In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow.

ON THE GEOMETRY OF THE CHARACTERISTIC CLASS OF A STAR PRODUCT ON A SYMPLECTIC MANIFOLD

The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.

Attractors and time averages for random maps

Considering random noise in finite dimensional parameterized families of diffeomorphisms of a compact finite dimensional boundaryless manifold M, we show the existence of time averages for almost every orbit of each point of M, imposing mild conditions on the families; see Section 2.4. Moreover these averages are given by a finite number of physical absolutely continuous stationary probability measures.We use this result to deduce that situations with infinitely many sinks and Hénon-like attractors are not stable under random perturbations, e.g., Newhouse’s and Colli’s phenomena in the generic unfolding of a quadratic homoclinic tangency by a one-parameter family of diffeomorphisms.

Gauge Dependence in the Theory of Non-Linear Spacetime Perturbations

Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we introduce the notion of knight diffeomorphism, that generalises the usual concept of flow, and prove a Taylor's formula for the action of a knight on a general tensor field. Then, we show that any one-parameter family of diffeomorphisms can be approximated by a family of suitable knights. Since in perturbation theory the gauge freedom is given by a one-parameter family of diffeomorphisms, the expansion of knights is used to derive our transformation formula. The problem of gauge dependence is a purely kinematical one, therefore our treatment is valid not only in general relativity, but in any spacetime theory.

Ω-stable limit set explosions

Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are Ω \Omega -stable.