Holomorphic Dynamics on Bounded Symmetric Domains of Finite Rank.

Abstract

In this thesis we present new results in holomorphic dynamics on rank-2 bounded symmetric domains, which can be infinite-dimensional. Some of these results have been published in [12]. Together with other current research, this establishes a comprehensive theory of the dynamics of fixed-point-free holomorphic self-maps on rank-2 bounded symmetric domains. Jordan theory is the novel approach used to achieve these results, which relates to the hyperbolic geometry of bounded symmetric domains. We examine the iterates of a fixed-point-free holomorphic self-map on the open unit balls D of two classes of JB*-triples: 1. A finite `1-sum V of Hilbert spaces; 2. The Banach space L(C2,H) of all bounded linear operators from C2 to a Hilbert space H. The main results in each case are an explicit description in Jordan theoretic terms of the invariant domains of f and an analysis of the subsequential limit points of the iterates of f in the topology of locally uniform convergence. Details are given as follows. Let f : D ! D be a compact fixed-point-free holomorphic map. We show the existence of horospheres S(⇠, #) at a boundary point ⇠ of D, parameterised by a positive number #, satisfying f(S(⇠, #) \ D) ⇢ S(⇠, #) \ D. These horospheres are described in terms of the Bergmann operator. 6 7 In Case 1, where V is a sum of p Hilbert spaces V1, . . . ,Vp, the horosphere S(⇠, #) at the boundary point ⇠ = (⇠1, . . . , ⇠p) has the form S(⇠, #) = Yp j=1 Sj(⇠j,#) where, for some nonempty subset J of {1, ..., p}, Sj(⇠j,#) = Dj for j 62 J and, for j 2 J, Sj(⇠j,#) = #2j ⇠j + B(#j⇠j,#j⇠j)1/2(Dj) where Dj is the open unit ball of Vj and #j > 0. In Case 2, the horosphere has the form S(⇠, #) = #21 e + #22 v + B (#1e + #2v, #1e + #2v)1/2 (D) where #1 2 (0, 1), #2 2 [0, 1) and e is a minimal tripotent. Leveraging these results we analyse the subsequential limit points of (fn). In Case 1, we prove that each limit point h of the iterates (fn) satisfies ⇠j 2 ⇡j & h(D) for all j 2 J and ⇡j&h(·) = ⇠j whenever ⇡j&h(D) meets the boundary of Dj , where ⇡j is the coordinate map (x1, ...,xp) 2 D 7! xj 2 Dj . In Case 2, the boundary point ⇠, takes the form e+%v,, where e is a minimal tripotent, % 2 [0, 1] and, if % 6= 0, v is a minimal tripotent. For each limit point h of (fn), we have h(D) ⇢ Ku for some tripotent u satisfying Ku \ Ke 6= ;, where Ka denotes the boundary component in D containing a.