Persistence of nonhyperbolic measures for <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-diffeomorphisms

In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.     

Analytic Classification of Fuchsian Singular Points

We study the local analytic classification of Fuchsian singular points. The resonance formal normal form (FNF) of a system with a Fuchsian singular point, as well as the local analytic equivalence of a system to its resonance FNF, is well known. However, there are distinct resonance FNFs locally analytically equivalent to each other. The main theorem of the paper reduces the problem of local analytic equivalence of resonance FNFs to a problem about conjugacy of certain matrices associated to two FNFs (which are nil-triangular) by a block upper triangular matrix. As a consequence, the local analytic classification of Fuchsian singular points reduces to the study of the orbits of the group of block upper triangular matrices on nil-triangular matrices by conjugation.     

Convergence of the Krylov-Bogolyubov procedure in Bowan’s example

In this paper, we study the behavior of time averages of the measure in Bowan’s example: a vector field on the plane with two saddles joined by two separatrix connections. We present an explicit criterion for the convergence of averaged measures and describe the set of their partial limits. As a consequence, we show that, for a typical initial measure, its time averages do not converge.     

Sur la dynamique unidimensionnelle en régularité intermédiaire

Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class (i.e. between C ¹ and C ²). In particular, we show some generalizations of Denjoy theorem and the classical Kopell lemma for abelian groups. These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class C ¹. We conclude with some remarks about the stationary measure.     

Random Conformal Dynamical Systems

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversally conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts a ball exponentially. We deduce some results about the unique ergodicity. Received: June 2005, Revision: January 2006, Accepted: March 2006     

Contraction of orbits in random dynamical systems on the circle

The paper deals with a theoretical justification of the effect, observed in computer experiments, of convergence of orbits (without tending to any particular point) in random dynamical systems on the circle. The corresponding theorem is proved under certain assumptions satisfied, in particular, in some C ¹-open domain in the space of random dynamical systems.It follows from this theorem that the corresponding skew product has two invariant measurable sections, naturally called an attractor and a repeller. Moreover, it turns out that convergence of orbits and the uniqueness of a stationary measure, phenomena that are mutually exclusive in the case of a single map, typically coexist in random dynamical systems.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 36–54, 2004Original Russian Text Copyright © by V. A. Kleptsyn and M. B. NalskiiSupported in part by RFBR grants 02-01-00482 and 02-01-22002 and CRDF grant RM1-2358-MO-02.     

Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets

Potential Analysis - We consider a family of dense Gδ subsets of [0, 1], defined as intersections of unions of small uniformly distributed intervals, and study their logarithmic capacity....     

Structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points

We study the structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points. This property generalizes the local expansivity property, and at present there are no known examples of minimal C ² actions of finitely generated groups of circle diffeomorphisms for which this generalized property does not hold.     

Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins

We prove that the set of smooth maps of an annulus with two attracting sets, both having metrically dense attracting basins, is open in an appropriate function space. Thus a 1994 claim of I. Kan is justified. To V. I. Arnold for more than half-centennial anniversary of his creative work, which in fluenced so many mathematicians, including the authors of this paper