Generic Diffeomorphisms Research Articles

Aperiodic chain recurrence classes of $C^{1}$-generic diffeomorphisms

Aperiodic chain recurrence classes of $C^{1}$-generic diffeomorphisms

A clarification on prevailing misconceptions in unimodular gravity

The traditional presentation of Unimodular Gravity (UG) consists on indicating that it is an alternative theory of gravity that restricts the generic diffeomorphism invariance of General Relativity. In particular, as often encountered in the literature, unlike General Relativity, Unimodular Gravity is invariant solely under volume-preserving diffeomorphisms. That characterization of UG has led to some confusion and incorrect statements in various treatments on the subject. For instance, sometimes it is claimed (mistakenly) that only spacetime metrics such that |det gμν | = 1 can be considered as valid solutions of the theory. Additionally, that same (incorrect) statement is often invoked to argue that some particular gauges (e.g. the Newtonian or synchronous gauge) are not allowed when dealing with cosmological perturbation theory in UG. The present article is devoted to clarify those and other misconceptions regarding the notion of diffeomorphism invariance, in general, and its usage in the context of UG, in particular.

-chain closing lemma for certain partially hyperbolic diffeomorphisms

Abstract For every $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f, if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y, there exist true orbits from U to V by arbitrarily $C^r$ -small perturbations. As a consequence, we prove that for $C^r$ -generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.

Eventual Fitting Shadowing Property for Hyperbolic Dynamical Systems

Let be a diffeomorphism map on a closed smooth manifold for dimension . We explain in this work any chain transitive set of generic diffeomorphism , if a diffeomorphism has another type of shadowing property is called, the eventual shadowing property on locally maximal chain transitive set, then is hyperbolic. In general, the eventual fitting shadowing property is not fulfilled in hyperbolic dynamical systems (satisfy in case L is Anosov diffeomorphism map) . In this paper, several concepts were presented. These concepts can be re-examined on other important spaces, and their impact on finding dynamical characteristics that may be employed in solving some mathematical problems.

Topological pressure for conservative C 1-diffeomorphisms with no dominated splitting

We prove three formulas for computing the topological pressure of C 1 -generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For C 1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and ϕ m ( x ) := − 1 m log ⁡ ‖ D x f m ‖ , m ∈ N , we show that there exist equilibrium states with zero entropy and there exists a transition point t 0 for the one parameter family { t ϕ m } t ≥ 0 , such that there is no equilibrium states for t ∈ [ 0 , t 0 ) and there is an equilibrium state for t ∈ [ t 0 , + ∞ ) .

Generic Equidistribution of Periodic Orbits for Area-Preserving Surface Maps

Abstract We prove that a $C^{\infty }$-generic area-preserving diffeomorphism of a closed, oriented surface admits a sequence of equidistributed periodic orbits. This is a quantitative refinement of the recently established generic density theorem for area-preserving surface diffeomorphisms. The proof has two ingredients. The first is a “Weyl law” for PFH spectral invariants, which was used to prove the generic density theorem. The second is a variational argument inspired by the work of Marques–Neves–Song and Irie on equidistribution results for minimal hypersurfaces and three-dimensional Reeb flows, respectively.

The three-dimensional generalized Hénon map: Bifurcations and attractors.

We study dynamics of a generic quadratic diffeomorphism, a 3D generalization of the planar Hénon map. Focusing on the dissipative, orientation preserving case, we give a comprehensive parameter study of codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include Hénon-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.

On the Shroer\u2013Sauer\u2013Ott\u2013Yorke Predictability Conjecture for Time-Delay Embeddings

Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if mu is a natural measure for a smooth diffeomorphism of a Riemannian manifold and k is greater than the information dimension of mu , then k time-delayed measurements of a one-dimensional observable h are generically sufficient for a predictable reconstruction of mu -almost every initial point of the original system. This reduces by half the number of required measurements, compared to the standard (deterministic) setup. We prove the conjecture for ergodic measures and show that it holds for a generic smooth diffeomorphism, if the information dimension is replaced by the Hausdorff one. To this aim, we prove a general version of predictable embedding theorem for injective Lipschitz maps on compact sets and arbitrary Borel probability measures. We also construct an example of a C^infty -smooth diffeomorphism with a natural measure, for which the conjecture does not hold in its original formulation.

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

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.

Asymptotic measure-expansiveness for generic diffeomorphisms

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 .

Symmetries of vector fields: The diffeomorphism centralizer

<p style='text-indent:20px;'>In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a <inline-formula><tex-math id="M1">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

Dirac Physical Measures on Saddle-Type Fixed Points

In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a $$C^1$$ generic diffeomorphism, a Dirac invariant measure whose statistical basin of attraction is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a $$C^1$$ -diffeomorphism having a Dirac invariant measure, supported on a saddle-type hyperbolic fixed point, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure. Our technique can be applied also to construct a $$C^1$$ diffeomorphism whose set of points with historic behaviour has positive measure and is nowhere dense.

A note on the definition of gravitational energy for quadratic curvature gravity via topological regularization* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11865006 and 11847152).

Within the framework of four-dimensional quadratic curvature gravities in the appearance of a negative cosmological constant, a definition for the gravitational energy of solutions with anti-de Sitter (AdS) asymptotics was put forward by Giribet et al. [Phys. Rev. D 98 044046 (2018)]. This was achieved by adding proper topological invariant terms to the gravity action to render the variation problem well-posed. We prove that the definition via the procedure of topological regularization can be covered by our previous work [Int. J. Mod. Phys. A 35 2050102 (2020)] in four dimensions. Motivated by this, we further generalize the results to generic diffeomorphism invariant theories of gravity in arbitrary even dimensions.

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

Abstract Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

Finiteness of partially hyperbolic attractors with one-dimensional center

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.

Ergodic Measures with Multi-zero Lyapunov Exponents Inside Homoclinic Classes

In this paper, we prove that for $$C^1$$-generic diffeomorphisms, if a homoclinic class contains periodic orbits of indices i and j with $$j>i+1$$, and the homoclinic class has no-domination of index l for any $$l\in \{i+1,\ldots ,j-1\}$$, then there exists a non-hyperbolic ergodic measure with more than one vanishing Lyapunov exponents and whose support is the whole homoclinic class. Some other results are also obtained.

Asymptotic orbital shadowing property for diffeomorphisms

Abstract Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

Saturation of generalized partially hyperbolic attractors

ABSTRACTWe prove the saturation of a generalized partially hyperbolic attractor of a map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a -generic diffeomorphism has zero volume. In contrast, by modification of the Poincaré cross section of Lorenz geometric model, we build a -diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.

Continuum-wise expansiveness for generic diffeomorphisms

Let M be a closed smooth manifold and let be a diffeomorphism. C1-generically, a continuum-wise expansive satisfies Axiom A without cycles. Let and let . There are a C1 neighborhood of and a residual set such that for any , g is not continuum-wise expansive, where is the set of all robustly transitive diffeomorphisms on

Vector fields satisfying the barycenter property

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].