Class Of Diffeomorphisms Research Articles

Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle

This paper introduces class G containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of 2q periodic points, where q is some natural number. So Cartesian products of two such diffeomorphisms has 4q1q2 periodic points where q1 corresponds to the first transformation and q2 corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains 2q1q2 saddle points, q1q2 sinks, and q1q2 sources; 4 points from mentioned 4q1q2 periodic ones are fixed, and the remaining 4q1q2−4 points have period 2. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism f of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over f. The authors introduce the concept of suspension over diffeomorphisms of class G, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold T2×[0,1]/φ, where φ:T2→T2. The main result of the paper says that suspensions over diffeomorphisms of the class G are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions ϕt and ϕ′t implies the topological conjugacy of ϕ and ϕ′.

Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a surface diffeomorphism

Abstract It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy type of the closure of the invariant manifold of a saddle point of a polar diffeomorphism of a 2-torus to any prescribed homotopically nontrivial type. The arc constructed in the process is stable and does not change the topological conjugacy class of the original diffeomorphism. The ideas that are proposed here for constructing such an arc for a 2-torus can naturally be generalized to surfaces of greater genus. Bibliography: 32 titles.

A hyperbolicity criterion for a class of diffeomorphisms of an infinite-dimensional torus

On an infinite-dimensional torus , where is an infinite-dimensional real Banach space and is an abstract integer lattice, a special class of diffeomorphisms is considered. It consists of the maps equal to sums of invertible bounded linear operators preserving and -smooth periodic additives. Necessary and sufficient conditions ensuring that such maps are hyperbolic (that is, are Anosov diffeomorphisms) are obtained. Bibliography: 15 titles.

Hamiltonian circle actions on complete intersections

We study the problem of determining which diffeomorphism classes of K\"{a}hler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.

Bavard's systolically extremal Klein bottles and three dimensional applications

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, we prove the existence on each diffeomorphism class of non-orientable Bieberbach 3-manifolds of a two-parameter family of singular Riemannian metrics that are systolically extremal in their conformal class. The proof uses a one-parameter family of singular Riemannian metrics on the Klein bottle discovered by C. Bavard ([3]): each one of these metrics is extremal in its conformal class.

Elements of hyperbolic theory on an infinite-dimensional torus

On the infinite-dimensional torus $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional Banach torus and $\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ is considered. It consists of the maps $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ whose differentials $DG$ and $D(G^{-1})$ are uniformly bounded and uniformly continuous on $\mathbb{T}^{\infty}$. For diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ elements of hyperbolic theory are presented systematically, starting with definitions and some auxiliary facts and ending by more advanced results. The latter include a criterion for hyperbolicity, a theorem on the $C^1$-roughness of hyperbolicity for diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$, the Hadamard-Perron theorem, as well as a fundamental result of hyperbolic theory, the fact that each Anosov diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ has a stable and an unstable invariant foliation. Bibliography: 34 titles.

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensonal Gorenstein contact lens spaces. As illustrative contact topology applications, we show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism.

Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

A new class of non-injective polynomial local diffeomorphisms on the plane

In this short note we provide the example with the lowest degree known so far of a non-injective local polynomial diffeomorphism F=(p,q):R2→R2. In our example p has degree 10 and q has degree 15, rather than 10 and 25, respectively, known up to now as the smallest degrees for the coordinates of F. Our construction was based on S. Pinchuk celebrated counterexample to the real Jacobian conjecture.

On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Boundedness questions for Calabi–Yau threefolds

In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2). The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).

Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

AbstractWe explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

The Classification of Diffeomorphism Classes of Real Bott Manifolds

A real Bott manifold (RBM) is obtained as the orbit space of the n-torus T^n by a free action of an elementary abelian 2-group ZZ_2^n. This paper deals with the classification of some particular types of RBMs of dimension n, so that we know the number of diffeomorphism classes in such RBMs.

Euler classes of vector bundles over manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

Johnson homomorphisms

Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S. Morita, are tools for understanding Torelli groups. This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman–Turaev Lie bialgebra, and the work of various authors on cohomological methods for determining the stable image of generalized Johnson homomorphisms. Various open problems and conjectures are included. Even though the Johnson homomorphisms were originally defined and studied by topologists, they are important in understanding arithmetic properties of mapping class groups and moduli spaces of curves. We define arithmetic Johnson homomorphisms, which extend the generalized Johnson homomorphisms, and explain how the Turaev cobracket constrains their images.

On a class of Anosov diffeomorphisms on the infinite-dimensional torus

Рассматривается некоторый вполне естественный класс диффеоморфизмов $G$, действующих из $\mathbb{T}^{\infty}$ в $\mathbb{T}^{\infty}$, где $\mathbb{T}^{\infty}$ - бесконечномерный тор (прямое произведение счетного числа окружностей с топологией равномерной покоординатной сходимости). Интересующие нас диффеоморфизмы допускают представление в виде суммы линейного гиперболического отображения и периодической добавки. Предлагается набор конструктивных достаточных условий, при которых любое отображение $G$ из нашего класса является гиперболическим, т. е. диффеоморфизмом Аносова на торе $\mathbb{T}^{\infty}$. Кроме этого, при выполнении упомянутых условий устанавливаются следующие стандартные факты из гиперболической теории: наличие устойчивого и неустойчивого инвариантных слоений, топологическая сопряженность с линейным гиперболическим автоморфизмом тора, структурная устойчивость $G$. Библиография: 21 наименование.

On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

The classifying space of the one‐dimensional bordism category and a cobordism model for TC of spaces

The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $\Omega^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bord_1^{red}(X)$ can be thought of as a cobordism model for the topological cyclic homology $TC(\mathbb{S}[\Omega X])$. In the second part of the paper we construct an infinite loop space map $B(hBord_1^{red}) \to Q(\Sigma^2 \mathbb{CP}^\infty_+)$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bord_1^{red}$ representing the Miller-Morita-Mumford classes $\kappa_i \in H^{2i+2}((B(hBord_1); \mathbb{Q})$.

On a class of Anosov diffeomorphisms on the infinite-dimensional torus

We study a quite natural class of diffeomorphisms on , where is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any in our class is hyperbolic, that is, an Anosov diffeomorphism on . Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of .

A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.