Class Of Diffeomorphisms Research Articles

Rough Diffeomorphisms with Basic Sets of Codimension One

The review is devoted to the exposition of results (including those of the authors of the review) obtained from the 2000s until the present, on topological classification of structurally stable cascades defined on a smooth closed manifold M n (n ≥ 3) assuming that their nonwandering sets either contain an orientable expanding (contracting) attractor (repeller) of codimension one or completely consist of basic sets of codimension one. The results presented here are a natural continuation of the topological classification of Anosov diffeomorphisms of codimension one. The review also reflects progress related to construction of the global Lyapunov function and the energy function for dynamical systems on manifolds (in particular, a construction of the energy function for structurally stable 3-cascades with a nonwandering set containing a two-dimensional expanding attractor is described).

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

A family of closed manifolds is said to be cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. Cohomological rigidity is established here for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. The class of 3-dimensional combinatorial simple polytopes different from tetrahedra and without facets forming 3- and 4-belts is studied. This class includes mathematical fullerenes, that is, simple 3- polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope in admits in Lobachevsky 3-space a right-angled realisation which is unique up to isometry. Our families of smooth manifolds are associated with polytopes in the class . The first family consists of 3-dimensional small covers of polytopes in , or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in . Our main result is that both families are cohomologically rigid, that is, two manifolds and from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if and are diffeomorphic, then their corresponding polytopes and are combinatorially equivalent. These results are intertwined with classical subjects in geometry and topology such as the combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds, and invariance of Pontryagin classes. The proofs use techniques of toric topology. Bibliography: 69 titles.

Hyperbolic annulus principle

We study a special class of diffeomorphisms of an annulus (the direct product of a ball in ℝ k , k ≥ 2, by an m-dimensional torus). We prove the so-called annulus principle; i.e., we suggest a set of sufficient conditions under which each diffeomorphism in a given class has an m-dimensional expanding hyperbolic attractor.

COMPLETE INTERSECTIONS WITH S 1-ACTION

We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S^1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also prove that in any odd complex dimension only finitely many complete intersections can carry a smooth effective action by a torus of rank $>1$.

The Myers-Steenrod theorem for Finsler manifolds of low regularity

We prove a version of Myers-Steenrod's theorem for Finsler manifolds under minimal regularity hypothesis. In particular we show that an isometry between $C^{k,\alpha}$-smooth (or partially smooth) Finsler metrics, with $k+\alpha>0$, $k\in \mathbb{N} \cup \{0\}$, and $0 \leq \alpha \leq 1$ is necessary a diffeomorphism of class $C^{k+1,\alpha}$. A generalisation of this result to the case of Finsler 1-quasiconformal mapping is given. The proofs are based on the reduction of the Finlserian problems to Riemannian ones with the help of the the Binet-Legendre metric.

Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of "stable" and "unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

Conjugacy classes of diffeomorphisms of the interval in $\mathcal {C}^{1}$-regularity

We consider the conjugacy classes of diffeomorphisms of the interval, endowed with the $C^1$-topology. Given two diffeomorphisms $f,g$ of $[0; 1]$ without hyperbolic fixed points, we give a complete answer to the following two questions: under what conditions does there exist a sequence of smooth conjugates $h_n f h_n^{-1}$ of $f$ tending to $g$ in the $C^1$-topology? under what conditions does there exist a continuous path of $C^1$-diffeomorphisms $h_t$ such that $h_t f h_t^{-1}$ tends to $g$ in the $C^1$-topology? We also present some consequences of these results to the study of $C^1$-centralizers for $C^1$-contractions of $[0;\infty)$; for instance, we exhibit a $C^1$-contraction whose centralizer is uncountable and abelian, but is not a flow.

Nonnegatively curved Euclidean submanifolds in codimension two

We provide a classification of compact Euclidean submanifolds M^n\subset\mathbb R^{n+2} with nonnegative sectional curvature, for n \ge 3 . The classification is in terms of the induced metric (including the diffeomorphism classification of the manifold), and we study the structure of the immersions as well. In particular, we provide the first known example of a nonorientable quotient (\mathbb S^{n-1} \times \mathbb S^1)/{\mathbb Z_2} \subset \mathbb R^{n+2} with nonnegative curvature. For the 3-dimensional case, we show that either the universal cover is isometric to \mathbb S^2 \times \mathbb R , or M^3 is diffeomorphic to a lens space, and the complement of the (nonempty) set of flat points is isometric to a twisted cylinder (N^2 \times \mathbb R)/\mathbb Z . As a consequence we conclude that, if the set of flat points is not too big, there exists a unique flat totally geodesic surface in M^3 whose complement is the union of one or two twisted cylinders over disks.

Family of measures on a space of curves that are quasi-invariant with respect to some action of diffeomorphisms group

A family of quasi-invariant measures on the special functional space of curves in a finite-dimensional Euclidean space with respect to the action of diffeomorphisms is constructed. The main result is an explicit expression for the Radon–Nikodym derivative of the transformed measure relative to the original one. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms. These measures can be used to obtain irreducible unitary representations of the diffeomorphisms group which will be studied in future research. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.

Groups of asymptotic diffeomorphisms

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the approach of V. Arnold [1].

Singularities of tangent surfaces to generic space curves

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities.

Magnetized Kerr/CFT correspondence

We extend the conjectured Kerr/CFT correspondence to the case of extremal Kerr black holes immersed by a magnetic field, namely the extremal Melvin–Kerr black holes. We compute the central charge which appears in the associated Virasoro algebra generated by a class of diffeomorphisms that satisfies a set of boundary conditions in the near horizon of an extremal Melvin–Kerr black hole. Our results support the Kerr/CFT conjecture, where the macroscopic Bekenstein–Hawking entropy for an extremal Melvin–Kerr black hole matches the result obtained from a dual 2D CFT microscopic computation using Cardy formula. Interestingly, the dual CFT description could be non-unitary, due to the possibility of negative central charge.

Anosov Diffeomorphisms and $${\gamma}$$ γ -Tilings

We consider a toral Anosov automorphism \({G_\gamma:{\mathbb{T}}_\gamma\to{\mathbb{T}}_\gamma}\) given by \({G_\gamma(x,y)=(ax+y,x)}\) in the \({ }\) base, where \({a\in\mathbb{N} \backslash\{1\}}\), \({\gamma=1/(a+1/(a+1/\ldots))}\), \({v=(\gamma,1)}\) and \({w=(-1,\gamma)}\) in the canonical base of \({{\mathbb{R}}^2}\) and \({{\mathbb{T}}_\gamma={\mathbb{R}}^2/(v{\mathbb{Z}} \times w{\mathbb{Z}})}\). We introduce the notion of \({\gamma}\)-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of \({G_\gamma}\); (ii) affine classes of \({\gamma}\)-tilings; and (iii) \({\gamma}\)-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.

The annulus principle in the existence problem for a hyperbolic strange attractor

A certain special class of diffeomorphisms of an `annulus' (equal to the Cartesian product of a ball in , , and a circle) is investigated. The so-called annulus principle is established, that is, a list of sufficient conditions ensuring that each diffeomorphism in this class has a strange hyperbolic attractor of Smale-Williams solenoid type is given. Bibliography: 20 titles.

Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms

We prove that, for any E u ⊕ E cs partially hyperbolic C 2 diffeomorphism, the ω-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the local generecity of the existence of a “thick” Milnor attractor in the class of boundary-preserving diffeomorphisms of the product of the interval and the 2-torus to this conjecture.

Cartan subalgebras of operator ideals

Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal B}({\mathcal H})$. A Cartan subalgebra ${\mathcal C}$ of ${\mathcal I}$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~${\mathcal I}$ and its conjugacy class is defined herein as the set of Cartan subalgebras $\{V{\mathcal C} V^*\mid V\in U_{\mathcal I}({\mathcal H})\}$. For nonzero proper ideals ${\mathcal I}$ we construct an uncountable family of Cartan subalgebras of ${\mathcal I}$ with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when ${\mathcal I}$ is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on~${\mathcal B}$. In the case when ${\mathcal I}$ is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of ${\mathcal I}$ become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{\mathcal I}({\mathcal H})$ on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of $U_{\mathcal I}({\mathcal H})$ and we give its construction.

On Stein fillings of contact torus bundles

We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein fillings of (Y,C).

Homology stability for symmetric diffeomorphism and mapping class groups

AbstractFor any smooth compact manifoldWwith boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set ofkpoints orkembedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms ofWconnected sum withkcopies of an arbitrary compact smooth manifoldQof the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

Morse bifurcations of transition states in bimolecular reactions

The transition states and dividing surfaces used to find rate constants for bimolecular reactions are shown to undergo Morse bifurcations, in which they change diffeomorphism class, and to exist for a large range of energies, not just immediately above the critical energy for first connection between reactants and products. Specifically, we consider capture between two molecules and the associated transition states for the case of non-zero angular momentum and general attitudes. The capture between an atom and a diatom, and then a general molecule are presented, providing concrete examples of Morse bifurcations of transition states and dividing surfaces.

Mixing-like properties for some generic and robust dynamics

We show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using the results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robustly topologically mixing. We also show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class.