On convergence theorems for homeomorphisms of Sobolev class

We prove a new convergence theorem for homeomorphisms of Sobolev class with a locally summable upper bound of deformations. This theorem allows us to generalize the known Strebel and Bers-Boyarskii convergence theorems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 249–259, February, 1995.     

On quasisymmetric homeomorphisms

We introduce and study some operators and functions induced by a quasisymmetric homeomorphism. By means of these operators and functions, we study when a quasisymmetric homeomorphism is symmetric or even belongs to the Weil-Petersson class.     

On quasimonotone increasing homeomorphisms

Let f ? C(\Bbb Rⁿ,\Bbb Rⁿ) f\in C(\Bbb R^n,\Bbb R^n) be quasimonotone increasing such that Y(f(y)-f(x)) ￡ -c Y(y-x) (x << y) \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) for a linear and strictly positive functional Y \Psi and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of x￠=f(x) x'=f(x) .     

On commutators of equivariant homeomorphisms

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing H₁(HG(M)) for more complicated G-manifolds.     

On critical circle homeomorphisms

We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a Denjoy theorem that such maps are conjugated to a rotation on the circle.Dedicated to the memory of R. MañéPartially supported by NSF grant DMS-9704368 and the Sloan Foundation.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

On the convergence of LePage series in Skorokhod space

We consider the problem of the convergence of the so-called LePage series in the Skorokhod space D^d=D([0,1],R^d) and provide a simple criterion based on the moments of the increments of the random process involved in the series. This provides a simple sufficient condition for the existence of an α-stable distribution on D^d with given spectral measure.     

On convergence on the boundary of the unit ball in dual space

In this paper some results that are known for extreme points of the unit ball in dual space are carried over to a more general case, namely to the case of the boundary of the ball ( B is the boundary of the unit ballB in the space dual toX if everyx X achieves its maximum value onB at some point of ). For example, it is established that if a set is bounded inX and countably compact in(X, ), then it is weakly compact inX.Translated fromMatematickeskie Zametki, Vol. 59, No. 5, pp. 753–758, May, 1996.     

On the forcing relation for surface homeomorphisms

Publications mathématiques de l'IHÉS -     

On the compactness of homeomorphisms with integrable dilatations

In this paper, a distortion theorem and some equicontinuity and compactness theorems are obtained for the homeomorphismf with the sphere dilatationH(x,f)∈L _loc(D). Supported by the National Natural Science Foundation of China and the Dctoral Foundation of the Education Commission of China     

On the weak isomorphism of strictly ergodic homeomorphisms

Two strictly ergodic homeomorphism (on an infinite dimensional torus) are constructed, each of which is a continuous homomorphic image of the other, but which are not measure-theoretically isomorphic.     

On fixed point sets of homeomorphisms of then-ball

Conditions are investigated under which a subsetA can be the fixed point set of a homeomorphism ofB ⁿ . If eitherA ∩ ?B ⁿ ≠ Ø andn arbitrary orA ∩ ?B ⁿ =Ø andn even it is necessary and sufficient thatA is non-empty and closed. IfA ∩ ?B ⁿ =Ø andn odd, conditions which are either necessary or sufficient (but not both) are given.