The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.
On a compact Riemannian manifold (Vm, g), we consider the second order positive operator , where − Δb is the Laplace-Beltrami operator and b is a Morse-Smale (MS) field, ɛ a small parameter. We study the measures which are the limits of the normalized first eigenfunctions of Lɛ as ɛ goes to zero.
In the case of a general MS field b, such a limit measure is the sum of a linear combination of Dirac measures located at the singular point of b and a linear combination of measures supported by the limit cycles of b.
When b is an MS-gradient vector field, we use a blow-up analysis to determine how the sequence concentrates on the critical point set. We prove that a critical point belongs to the support of a limit measure only if the Topological Pressure defined by a variational problem (see [Kifer Y.: Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J. Math. 70 (1990), 1–47]) is achieved there. Also if a sequence converges to a measure in such a way that every critical points is a limit point of global maxima of the eigenfunction, then we can compute the weight of a limit measure. This result provides a link between the limits of the first eigenvalues and the associated eigenfunctions. We give an interpretation of this result in terms of the movement of a Brownian particle driven by a field and subjected to a potential well, in the small noise limit.
We are working with two topological notions of similarity of functions. We show, that these notions can be used to investigate some important properties of functions. Some types of generalized continuity are investigated. New optimization results are presented, too.
Let M be a complete Riemann manifold with dimension m and metric g. For p, q ∈ M and ℓ > 0, let the index I (g, p, q, ℓ) be the number of g-geodesics of length ℓ that join p to q. The following generic bounds for this index are the main results we present here. We denote by ℛ the space of complete Riemann metrics on M.
(a) For each p ∈ M, there is a residual 𝒢 (p) ⊂ ℛ such that for all g ∈ 𝒢(p)
(b) If M is compact, there is a residual 𝒢 ⊂ ℛ such that for all g ∈ 𝒢
These finiteness results are part of our study of the focal decomposition—i.e., the partition
Stability of this focal deomposition (as g varies) has a natural meaning, in analogy with structural stability in the theory of dynamical systems, and here we begin an investigation in that direction. Our methods involve the multi-transversality theory of J. Mather and the Bumpy Metric Theorem of R. Abraham, as proved by D. Anosov.
We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant ε goes to zero. If the drift of the diffusion is given by a Morse–Smale vector field b, the limits of the eigenfunctions concentrate on the recurrent set of b. A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set.
We consider generalized Morse–Smale vector fields, the recurrent set of which is composed of hyperbolic critical points, limit cycles and two dimensional torii. Under some compatibility conditions between the flow of b and the Riemannian metric g along each of these components, we prove that the support of a limit lies on those recurrent components of maximal dimension, where the topological pressure is achieved. Also, the restriction of the limit measure to either a cycle or a torus is absolutely continuous with respect to the unique invariant probability measure of the restriction of b to the cycle or the torus. When the torii are not charged, the restriction of the limit measure is absolutely continuous with respect to the arclength on the cycle and we have determined the corresponding density. Finally, the support of the limit measures and the support of the measures selected by the variational formulation of the topological pressure (TP) are identical.