scholarly journals A quantitative version of the Kupka-Smale theorem

1985 ◽  
Vol 5 (3) ◽  
pp. 449-472 ◽  
Author(s):  
Y. Yomdin
Keyword(s):  
Periodic Point ◽  
Smooth Manifold ◽  
Quantitative Version

AbstractLet Mm be a compact, m-dimensional smooth manifold. The n-periodic point x of a diffeomorphism f: M → M is called γ-hyperbolic, for γ≥O, if the eigenvalues λj, of dfn(x) satisfy . We prove that any Ck-diffeomorphism f: M → M, k≥3, for any ε>0 can be ε-approximated in Ck-norm by fe: M → M such that for any n each n-periodic point of fe is (a(ε))nα - hyperbolic. Here and ao>0 depends on f

scholarly journals Asymptotic measure-expansiveness for generic diffeomorphisms

2021 ◽  
Vol 19 (1) ◽  
pp. 470-476
Author(s):  
Manseob Lee
Keyword(s):  
Periodic Point ◽  
Smooth Manifold ◽  
Homoclinic Class ◽  
Axiom A

Abstract In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

scholarly journals Fast growth of the number of periodic points arising from heterodimensional connections

2021 ◽  
Vol 157 (9) ◽  
pp. 1899-1963
Author(s):  
Masayuki Asaoka ◽  
Katsutoshi Shinohara ◽  
Dmitry Turaev
Keyword(s):  
Periodic Point ◽  
Small Perturbation ◽  
Exponential Growth ◽  
Smooth Manifold ◽  
Small Neighborhood ◽  
Periodic Points ◽  
Natural Conditions ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Heteroclinic Points

We consider $C^{r}$ -diffeomorphisms ( $1 \leq r \leq +\infty$ ) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$ -small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$ , provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$ -generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

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