多复变数的Schwarz导数(Ⅴ)

本文从Thurston的观点出发,用二阶逼近来定义与讨论矩阵空间C~(m×n)（m≤n）中的域上全纯映照的Schwarz导数及高阶Schwarz导数,证明：如果它们存在的话,那么它们是在R_I（m,n）的紧对偶空间CG（m,n）的全纯自同构群下的相似不变量．并证明：这样得到的Schwarz导数与前几文［1－4］中由Ahlfors的观点得到的Schwarz导数是相一致的．此外,还应用这种观点定义与讨论了C~N中的域上全纯映照的Schwarz导数．     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

Universal functions for composition operators with non-automorphic symbol

For sequences (φ _n ) of eventually injective holomorphic self-maps of planar domains Ω, we present necessary and sufficient conditions for the existence of holomorphic functions f on whose orbits under the action of (φ _n ) are dense in H (Ω). It is deduced that finitely connected, but non-simply connected domains never admit such universal functions. On the other hand, if we allow arbitrary sequences of holomorphic self-maps (φ _n ), the situation changes dramatically.     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

The Schwarzian derivative in several complex variables (III)

In the point view of Lie group, the cross ratio and Schwarzian derivative in Cⁿ are defined and discussed, especially the Schwarzian derivative of holomorphic mappings on the domains in matrix space C^{m x n} is defined and discussed. It is proved that it is invariant up to similarity under the group of holomorphic automorphism of the Grassmann manifold CG(m, n). And it is also proved that the Schwarzian derivative equals zero if and only if the mapping is liaearly fractional. Project supported by the National Natural Science Foundation of China.     

Higher-Order Schwarz-Pick Estimates for Holomorphic Self-mappings on Classical Domains

In this paper,Schwarz-Pick estimates for high order Fréchet derivatives of holomorphic self-mappings on classical domains are presented.Moreover,the obtained result can deduce the early work on Schwarz...     

Applications holomorphes propres entre certains domaines bornés de ℂn

In this Note, we developp a new technic to study the existence of proper holomorphic mappings between a strictly pseudoconvex domain and certain special non-regular domains in ℂⁿ. In particular, it can be applied in the case of the minimal ball and the Lie ball. We prove that self-proper holomorphic mappings between such domains are biholomorphic. Furthermore, we establish a necessary and sufficient condition to factorize a proper holomorphic mapping by automorphisms. Scaling method is applied for the first time in a singular points of the boundary of the domains.     

多圆柱上Dirichlet空间中Toeplitz算子的交换性

本文讨论了多圆柱上Dirichlet空间中的正规Toeplitz算子以及全纯指标和反全纯指标的两个Toeplitz算子的交换性.我们证明具有多圆柱上Dirichlet空间中反全纯指标的两个Toeplitz算子可交换当且仅当两个指标是线性相关的,同时证明全纯指标和反全纯指标的两个Toeplitz算子可交换当且仅当两个指标有一个是常数.     

On close to linear cocycles

If we have a flow and a cocycle on this flow, , then is called close to linear if can be written as the direct sum of a linear (constant) cocycle and a cocycle in the closure of the coboundaries. Many of the desirable consequences of linearity hold for such cocycles and, in fact, a close to linear cocycle is cohomologous to a cocycle which is norm close to a linear one. Furthermore in the uniquely ergodic case all cocycles are close to linear. We also establish that a close to linear cocycle which is covering is cohomologous to one with the special property that it can be extended by piecewise linearity to an invertible cocycle from to itself. This implies that a suspension obtained from a close to linear cocycle is isomorphic to a time change of the suspension obtained from the identity cocycle.

     

SCHWARZ-PICK ESTIMATES FOR BOUNDED HOLOMORPHIC FUNCTIONS ON CLASSICAL DOMAINS

In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.     

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

关于逆紧全纯映射刚性的一个注记

本文对某类广义Hartogs三角形上的逆紧全纯自映射证明了刚性定理,即逆紧全纯自映射必定为全纯自同构．同时完全刻画了其全纯自同构群,并且给出了关于其全纯自同构以及两个这类域之间逆紧全纯映射的分类。     

Uniform Convergence of Schr?dinger Cocycles over Bounded Toeplitz Subshift

For locally constant cocycle defined on an aperiodic subshift, Damanik and Lenz proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. For any simple Toeplitz subshift, we proved that the corresponding Schr?dinger cocycle is uniform, although it does not satisfy condition (B) in general. In this paper, we study bounded Toeplitz subshift. In general, it does not satisfy condition (B); and it contains non-simple case, which make us cannot use Chebishev polynomial. By a combination of trace formula and avalanche principle, we prove that for any bounded Toeplitz subshift, the corresponding Schr?dinger cocycle is also uniform.     

THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.     

Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special class of bounded (1, 2, . . . , n)-balanced domains in \mathbb Cⁿ,{\mathbb C^n,} called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groups of these domains.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

Ergodic reduction of random products of two-by-two matrices

We consider a random product of two-by-two matrices of determinant one over an abstract dynamical system. When the two Lyapunov exponents are distinct, Oseledets’ theorem asserts that the matrix cocycle is cohomologous to a diagonal matrix cocycle. When they are equal, we show that the cocycle is conjugate to one of three cases: a rotation matrix cocycle, an upper triangular matrix cocycle, or a diagonal matrix cocycle modulo a rotation by π/2.