Properties of Fixed Point Sets and a Characterization of the Ball in ${\Bbb C}^n$

Isolated fixed point sets for holomorphic maps

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed.

Weighted Composition Operators from $F(p,q,s)$ to Bloch Type Spaces on the Unit Ball

Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B$ and $ψ(z)$ a holomorphic function on $B$, where $B$ is the unit ball of ${\Bbbb C}^n$. Let $0<p,s<+\infty, -n-1<q<+\infty, q+s>-1$ and $α\geq 0,$ this paper gives some necessary and sufficient conditions for the weighted composition operator $\wco$ induced by $ϕ$ and $ψ$ to be bounded and compact between the space $\Fs$ and $α$-{\sl Bloch} space $β^α.$

Interpolation sets for Hardy-Sobolev spaces on the boundary of the unit ball of ${\bf C}^n$

We study the interpolation sets for the Hardy-Sobolev spaces defined on the unit ball of ${\bf C}^n$. We begin by giving a natural extension to ${\bf C}^n$ of the condition that is known to be necessay and sufficient for interpolation sets lying on the boundary of the unit disc. We show that under this condition the restriction of a function in the Hardy-Sobolev space to the set always exists, and lies in a Besov space. We then show that under the assumption that there is an holomorphic distance function for the set, there is an extension operator from these spaces to the Hardy-Sobolev ones.

Fixed points and Determining Sets for Holomorphic Self-Maps of a Hyperbolic Manifold

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different.

Zero sets of holomorphic functions in the unit ball: non-radial growth characteristics

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint $|f|\leq \exp M$ on $\mathbb B$, where $M\not\equiv \pm \infty$ is $δ$-subharmonic function on $\mathbb B$ with Riesz charge $μ_M$. We give a scale of integral uniform constraints from above on the distribution of the set ${\sf Z}$ via the charge $ν_M$ in terms of $(2n-2)$-Hausdorff measure of the set $\sf Z$, as well as test convex radial functions and $ρ$-subsp…

Fuchsian Groups, Quasiconformal Groups, and Conical Limit Sets

We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is…

On vector-valued automorphic forms on bounded symmetric domains

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of the norms (as the weight goes to infinity) for a class of totally real submanifolds. We give an example of a CR submanifold of the ball, for which the norms of the associated automorphic forms have a different asymptotic behavior.

Approximation in the closed unit ball

In this expository article, we present a number of classic theorems that serve to identify the closure in the sup-norm of various sets of Blaschke products, inner functions and their quotients, as well as the closure of the convex hulls of these sets. The results presented include theorems of Carathéodory, Fisher, Helson-Sarason, Frostman, Adamjan-Arov-Krein, Douglas-Rudin and Marshall. As an application of some of these ideas, we obtain a simple proof of the Berger-Stampfli spectral mapping theorem for the numerical range of an operator.

$\Bbb{C}^{2}$上多项式自同构的随机动力系统

Random dynamical systems of polynomial automorphisms on $\Bbb{C}^{2}$

研究$\Bbb{C}^{2}$上多项式自同构（复广义Hénon映射及其共轭）的随机动力系统，证明一般系统具有“平均稳定性”，并推导出负Lyapunov指数、有限吸引极小集等性质，这些在确定性迭代中不成立。

Interpolating sequences for weighted Bergman spaces of the ball

Let $B_α^{p}$ be the space of $f$ holomorphic in the unit ball of $\Bbb C^n$ such that $(1-|z|^2)^αf(z) \in L^p$, where $0<p\leq\infty$, $α\geq -1/p$ (weighted Bergman space). In this paper we study the interpolating sequences for various $B_α^{p}$. The limiting cases $α=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and $A^{-α}$, the holomorphic functions with polynomial growth of order $α$, which have generated particular interest. In §1 we first collect some definitions and well-known facts about weighted Bergman spaces and then introduce the natural interpolation problem, alo…

Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball

We prove that to every connected Abelian subgroup $H$ of the biholomorphisms of the unit ball $\mathbb{B}^n$ we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative $C^*$-algebra on every weighted Bergman space. These symbols are of the form $a(z) = f(μ^H(z))$, where $μ^H$ is the moment map for the action of $H$ on $\mathbb{B}^n$. We show that, for this construction, if $H$ is a maximal Abelian subgroup, then the symbols introduced are precisely the $H$-invariant symbols. We provide the explicit computation of moment maps to obtain special sets o…

近球域在伯格曼球中的等周不等式

Isoperimetric inequality for nearly spherical domains in the Bergman ball

本文在伯格曼球中研究近球域的等周不等式，证明了相关定理并首次揭示了该领域的等周现象。

On fixed points of self maps of the free ball

In this paper, we study the structure of the fixed point sets of noncommutative self maps of the free ball. We show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace. We provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete Pick spaces.

再生核希尔伯特空间之间的距离与单位球中有限集的几何

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

本文研究再生核希尔伯特空间、乘子代数与点集几何的关系，引入适用于两者的距离变体，证明在Drury-Arveson空间有限维商空间中，空间接近等价于乘子代数接近及点集几乎全等。