• Login
    JavaScript is disabled for your browser. Some features of this site may not work without it.
    Holomorphic Dynamics on Bounded Symmetric Domains of Finite Rank. 
    •   QMRO Home
    • Queen Mary University of London Theses
    • Theses
    • Holomorphic Dynamics on Bounded Symmetric Domains of Finite Rank.
    •   QMRO Home
    • Queen Mary University of London Theses
    • Theses
    • Holomorphic Dynamics on Bounded Symmetric Domains of Finite Rank.
    ‌
    ‌

    Browse

    All of QMROCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects
    ‌
    ‌

    Administrators only

    Login
    ‌
    ‌

    Statistics

    Most Popular ItemsStatistics by CountryMost Popular Authors

    Holomorphic Dynamics on Bounded Symmetric Domains of Finite Rank.

    View/Open
    Rigby_Jeffrey_PhD_140815.pdf (7.175Mb)
    Publisher
    Queen Mary University of London
    Metadata
    Show full item record
    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.
    Authors
    Rigby, Jeffrey Michael
    URI
    http://qmro.qmul.ac.uk/xmlui/handle/123456789/13118
    Collections
    • Theses [3360]
    Copyright statements
    The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author
    Twitter iconFollow QMUL on Twitter
    Twitter iconFollow QM Research
    Online on twitter
    Facebook iconLike us on Facebook
    • Site Map
    • Privacy and cookies
    • Disclaimer
    • Accessibility
    • Contacts
    • Intranet
    • Current students

    Modern Slavery Statement

    Queen Mary University of London
    Mile End Road
    London E1 4NS
    Tel: +44 (0)20 7882 5555

    © Queen Mary University of London.