Title:

Holomorphic dynamics on bounded symmetric domains of finite rank

In this thesis we present new results in holomorphic dynamics on rank2 bounded symmetric domains, which can be infinitedimensional. Some of these results have been published in [12]. Together with other current research, this establishes a comprehensive theory of the dynamics of fixedpointfree holomorphic selfmaps on rank2 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 fixedpointfree holomorphic selfmap on the open unit balls D of two classes of JB*triples: 1. A finite `1sum 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 fixedpointfree 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.
