The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

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).     

Robust rigidity for circle diffeomorphisms with singularities

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C ¹-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C ¹-smoothly conjugate.     

Morse-Smale diffeomorphisms with three fixed points

It is proved that the closures of separatrices for a Morse-Smale diffeomorphism with three fixed points are flatly embedded spheres if the dimension of the manifold is at least 6 and may be wildly embedded spheres if the dimension of the manifold is 4.     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

We prove that any C^3+β-smooth diffeomorphism preserving the orientation of a circle with rotation number from the Diophantine class D_δ, 0 < β < δ < 1, is C^2+β−δ-smoothly conjugate to a rigid rotation of the circle by a certain angle. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 268–282, February, 2008.     

The Baire set fixing property and uniform continuity in topological groups

Every topological group that satisfies theM-Baire set fixing property and is left separable isdense in a retract of a product of metrizable groups. Since compact groups satisfy MBSFP, it is a corollary that compact groups are dyadic spaces (Kuz'minov). IfG satisfies MBSFP, then the left uniformity is separable if and only ifG satisfies the countable chain condition. Dense subgroups of left separable MBSFP groups satisfy MBSFP, and any product of left separable MBSFP groups satisfies MBSFP. A left separable group satisfies MBSFP if and only if the left uniformity is weak generated by natural maps to metrizable quotient groups. If for each ,A is a denseC-embedded set in the left separable MBSFP groupG , then the real-compactification A = A . It is a corollary that the product of pseudocompact groups is pseudocompact (Comfort and Ross). A locally compact group satisfies MBSFP if and only if it is metrizable or sigma compact (Ross and Stromberg, forG Abelian).     

Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

We study translation surfaces with rich groups of affine diffeomorphisms—“prelattice” surfaces. These include the lattice translation surfaces studied by W. Veech. We show that there exist prelattice but nonlattice translation surfaces. We characterize arithmetic surfaces among prelattice surfaces by the infinite cardinality of their set of points periodic under affine diffeomorphisms. We give examples of translation surfaces whose periodic points and Weierstrass points coincide.