Complex dynamics in higher dimensions.
Robertson, John Wesley
2000
Abstract
This thesis is on the complex dynamics of holomorphic maps f:<blkbd>Pr </blkbd>→<blkbd>Pr </blkbd> of complex projective space. We study the history space (studied extensively by Lyubich and Minsky in the <blkbd>P1</blkbd> case), introduce and study Fatou maps, study topological conjugacy and give new examples of critically finite maps. We study the complex analytic pieces which the history space <a><ac><blkbd>P</blkbd></ac><ac>&d4;</ac></a>r of <italic>f</italic> contains by considering natural holomorphic maps from an arbitrary reduced complex space <italic>Z into</italic> the history space. Any such map gives rise to a normal family of maps from <italic>Z</italic> into <blkbd>Pr</blkbd> (as follows from a theorem of Ueda). We study <sub>r</sub> dimensional manifolds which lie in the history space and show that the global unstable set of a hyperbolic fixed point <italic>p</italic> corresponds to a manifold <italic> M</italic> in the history space. If <italic>f</italic> is locally analytically linearizable at <italic>p</italic> then M=<blkbd>Cn</blkbd> . A Fatou map is a holomorphic map g:Z→<blkbd> Pr</blkbd> such that the sequence of forward iterates f&j0;i&j0; g</fen>i is a normal family. We refine work of Hubbard & Papadopol, Fornaess & Sibony and Ueda using precisely their methods but in the context of Fatou maps to show that g:Z→<blkbd> Pr</blkbd> is a Fatou map iff there is a lift <a><ac>g</ac><ac>&d4;</ac></a> :<a><ac>Z</ac><ac>&d5;</ac></a>→ <blkbd>Cr+1</blkbd>\ 0</fen> (where <italic>Z˜</italic> is the universal cover of <italic> Z</italic>) of <italic>g</italic> which lands in the zero set of the Green's function. We show that the set of Fatou maps from an arbitrary reduced complex space <italic>Z</italic> is compact. We derive a criterion for a Fatou component to be taut. It follows using work of Fornaess & Sibony that any recurrent Fatou component in <blkbd>P2</blkbd> which is not a Siegel domain is taut. We also study the stable manifold of a hyperbolic fixed point. We prove that if <italic>f</italic><sub>1</sub> and <italic>f</italic><sub> 2</sub> are holomorphic self maps of <blkbd>Pr</blkbd> which are <italic>topologically</italic> conjugate by an appropriate homeomorphism <italic>h</italic> of <blkbd>Pr</blkbd> then <italic>h</italic> lifts canonically to a topological conjugacy <italic> H</italic> between the corresponding self maps <italic>F</italic><sub>1</sub> and <italic>F</italic><sub>2</sub> of <blkbd>Cr+1</blkbd> . Moreover G2&j0;H=G1 where <italic>G</italic><sub>1</sub> and <italic>G</italic><sub>2 </sub> are the Green's functions of <italic>f</italic><sub>1</sub> and <italic> f</italic><sub>2</sub>. Lastly we present new examples of critically finite self maps of <blkbd>Pr</blkbd> .Subjects
Complex Space Complex Spaces Dimensions Dynamics Fatou Maps Higher Holomorphic Mappings
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