Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of \(d\) -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.     

Residual subsets in the space of finitely generated groups of diffeomorphisms of the circle

Finitely generated groups of diffeomorphisms of the circle are considered in different problems of geometry, wave theory, variational calculus, etc. In particular, the set of such groups contains groups freely acting on the orbits of almost each point of the circle. The present paper deals with the structure of the set of finitely generated groups of diffeomorphisms with a given number of generators and with the above property. It is shown that such a set contains a residual subset (i.e., contains the countable intersection of open everywhere dense subsets).     

Rigidity result on conjugacies of families of diffeomorphisms

Embedding flows are used to obtain a rigidity result on strongly topological conjugacy of families of diffeomorphisms, i.e. families of C^r(2?r?∞) diffeomorphisms, the strongly topologically conjugating homeomorphisms near degenerate saddle-nodes will be differentiable on center manifolds of the saddle-nodes.     

On a Version of the Hyperbolic Annulus Principle

A sufficiently general class of diffeomorphisms of the annulus (the direct product of a ball in \(\mathbb{R}^{k}\), k ≥ 2, by an m-dimensional torus) is studied. The so-called annulus principle, i.e., a set of sufficient conditions under which the diffeomorphisms of the class under study have a mixing hyperbolic attractor, is obtained.     

Commutators of contactomorphisms

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.     

On Algebraic Anosov Diffeomorphisms on Nilmanifolds

This article is devoted to the algebraic approaches to Anosov diffeomorphisms. All examples of Anosov diffeomorphisms known so far are connected directly or indirectly with compact nilmanifolds. We consider some new necessary conditions for existence of these diffeomorphisms on nilmanifolds. We demonstrated the absence of Anosov diffeomorphisms on some classes of nilmanifolds. We also prove some results on Lie groups and lattices in them.     

On Morse--Smale Diffeomorphisms with Four Periodic Points on Closed Orientable Manifolds

We study Morse--Smale diffeomorphisms of n-manifolds with four periodic points which are the only periodic points. We prove that for n= 3 these diffeomorphisms are gradient-like and define a class of diffeomorphisms inevitably possessing a nonclosed heteroclinic curve. For n 4, we construct a complete conjugacy invariant in the class of diffeomorphisms with a single saddle of codimension one.     

A special ergodic theorem for Anosov diffeomorphisms on the 2-torus

A strengthened version of the Birkhoff ergodic theorem for linear Anosov diffeomorphisms on the 2-torus is presented. Namely, it is proved that the Hausdorff dimension of the set of points at which partial limits of the time average strongly (by a constant) differ from the space average is strictly less than the dimension of the space (i.e., than 2).     

Some aspects of analysis on almost complex manifolds with boundary

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds, and the elliptic regularity of some diffeomorphisms of almost complex manifolds with boundary. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 47, Complex Analysis, 2007.     

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

New partially hyperbolic dynamical systems I

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.     

The closure of planar diffeomorphisms in Sobolev spaces

We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Müller and Spector.     

Many-Dimensional Morse–Smale Diffeomeophisms with a Dominant Saddle

Mathematical Notes - The article defines a class of Morse–Smale diffeomorphisms with a dominant saddle and provides necessary and sufficient conjugacy conditions for such diffeomorphisms. We...     

Horizontal curves of hyperbolic metrics

We analyse the fine convergence properties of one parameter families of hyperbolic metrics that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms. Such families arise naturally in the study of general curves of metrics on surfaces, and in one of the gradients flows for the harmonic map energy.     

Theoretical aspects of information security

In this work, the authors obtain a topological classification of the simplest diffeomorphisms of the sphere S ² having exactly one orbit of one-sided heteroclinic tangency. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Abelian extensions via prequantization

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω, to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extension of a simply connected Lie group can be obtained as the pull-back of such a prequantization abelian extension.     

Breakdown of symmetry in reversible systems

We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractor-repeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms).     

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability (probability one with respect to every invariant probability measure) are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.