Iteration of self-maps on a product of Hilbert balls

Let D=D1×⋯×Dp be a product of Hilbert balls, with coordinate maps πj:D¯→D¯j on the closure D¯, for j=1,…,p. Let f be a fixed-point free self-map on D, which is nonexpansive in the Kobayashi distance, and compact for p⩾2. We describe the horospheres invariant under f and show that there exist a bound...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 411; no. 2; pp. 773 - 786
Main Authors Chu, Cho-Ho, Rigby, Michael
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.03.2014
Subjects
Hilbert ball
Iteration
JB⁎-triple
Kobayashi distance
Polydisc
Kobayashi distance
Iteration
Polydisc
JB⁎-triple
Hilbert ball
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Summary:Let D=D1×⋯×Dp be a product of Hilbert balls, with coordinate maps πj:D¯→D¯j on the closure D¯, for j=1,…,p. Let f be a fixed-point free self-map on D, which is nonexpansive in the Kobayashi distance, and compact for p⩾2. We describe the horospheres invariant under f and show that there exist a boundary point (ξ1,…,ξp) of D and a nonempty set J⊂{1,…,p} such that each limit function h of the iterates (fn) satisfies ξj∈πj∘h(D)¯ for all j∈J and πj∘h(⋅)=ξj whenever πj∘h(D) meets the boundary of Dj. For a single Hilbert ball D1, either liminfn→∞‖f2n(0)‖<1 or (fn) converges locally uniformly to a constant map taking value at the boundary of D1.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.10.009