Iteration of holomorphic maps on Lie balls

We investigate iterations of fixed-point free holomorphic self-maps on a Lie ball of any dimension, where a Lie ball is a bounded symmetric domain and the open unit ball of a spin factor which can be infinite dimensional. We describe the invariant domains of a holomorphic self-map f on a Lie ball D when f   is fixed-point free and compact, and show that each limit function of the iterates (fⁿ)$(f^n)$ has values in a one-dimensional disc on the boundary of D  . We show that the Möbius transformation _ga$g_a$ induced by a nonzero element a in D may fail the Denjoy–Wolff-type theorem, even in finite dimension. We determine those which satisfy the theorem.     

Condition R and proper holomorphic maps between equidimensional product domains

We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as a product of proper mappings between the factor domains.     

Spherical space forms,CR mappings,and proper maps between balls

We determine precisely for which spherical space forms there are nontrivial smooth CR mappings to spheres. Equivalently we determine for which fixed point free finite unitary groups ? there exists a ?-invariant proper holomorphic rational map between balls. The answer is that the group must be cyclic and essentially only two classes of representations can occur. For these there are invariant polynomial examples.     

On boundary regularity of proper holomorphic mappings

We show that a proper holomorphic mapping from a domain with real-analytic boundary to a domain with real-algebraic boundary extends holomorphically to a neighborhood of . Received: 14 March 2001 / Published online: 1 February 2002     

On proper holomorphic mappings of strictly pseudoconvex domains

Siberian Mathematical Journal -     

A sharp bound for the degree of proper monomial mappings between balls

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.