Asymptotic measure-expansiveness for generic diffeomorphisms

Manseob Lee

doi:10.1515/math-2021-0037

Asymptotic measure-expansiveness for generic diffeomorphisms

Open Mathematics ◽

10.1515/math-2021-0037 ◽

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 .

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Measure-Expansive Homoclinic Classes for C1 Generic Flows

Mathematics ◽

10.3390/math8081232 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1232

Author(s):

Manseob Lee

Keyword(s):

Vector Field ◽

Smooth Manifold ◽

Closed Orbit ◽

Homoclinic Class ◽

Homoclinic Classes

In this paper, we prove that for a generically C1 vector field X of a compact smooth manifold M, if a homoclinic class H(γ,X) which contains a hyperbolic closed orbit γ is measure expansive for X then H(γ,X) is hyperbolic.

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A quantitative version of the Kupka-Smale theorem

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003072 ◽

1985 ◽

Vol 5 (3) ◽

pp. 449-472 ◽

Cited By ~ 5

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

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

Compositio Mathematica ◽

10.1112/s0010437x21007405 ◽

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.

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Banach Lie Algebroids

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-011-0035-y ◽

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.

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Missing the point in noncommutative geometry

Synthese ◽

10.1007/s11229-020-02998-1 ◽

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.

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On uniformly distributed orbits of certain horocycle flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001474 ◽

1982 ◽

Vol 2 (2) ◽

pp. 139-158 ◽

Cited By ~ 18

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).

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Shelah's work on non-semi-proper iterations, II

Journal of Symbolic Logic ◽

10.2307/2694981 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1865-1883 ◽

Cited By ~ 3

Author(s):

Chaz Schlindwein

Keyword(s):

Closed Subset ◽

Stationary Subset ◽

Finite Support ◽

Countable Support ◽

Final Model ◽

Axiom A ◽

Mahlo Cardinal ◽

The One ◽

Image Position ◽

Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

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Pseudoinversion of degenerate metrics

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203301309 ◽

2003 ◽

Vol 2003 (55) ◽

pp. 3479-3501 ◽

Cited By ~ 9

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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