Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008
Springer Science & Business Media, Jul 31, 2010 - Mathematics - 348 pages
The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector ?eld) acting on it. The study of a holomorphic dynamical system consists in describing the asymptotic behavior of the system, associating it with some invariant objects (easy to compute) which describe the dynamics and classify the possible holomorphic dynamical systems supported by a given manifold. The behavior of a holomorphic dynamical system is pretty much related to the geometry of the ambient manifold (for instance, - perbolic manifolds do no admit chaotic behavior, while projective manifolds have a variety of different chaotic pictures). The techniques used to tackle such pr- lems are of variouskinds: complexanalysis, methodsof real analysis, pluripotential theory, algebraic geometry, differential geometry, topology. To cover all the possible points of view of the subject in a unique occasion has become almost impossible, and the CIME session in Cetraro on Holomorphic Dynamical Systems was not an exception.
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algebraic degree analytic set analytic subset assume automorphisms basin bidegree blowup boundary bounded canonical bundle classiﬁcation closed p,p)-current complex manifold connected constant contains converge coordinates ddcu deduce deﬁned Deﬁnition denote disc dynamical systems eigenvalues endomorphism entire functions entropy equal equilibrium measure ergodic exponential Fatou component Fatou set ﬁber ﬁnite ﬁrst ﬁxed point foliation following result Hausdorff dimension Hence Holder continuous holomorphic function holomorphic local dynamical holomorphic map holonomy hyperbolic identity implies inﬁnite intersection isomorphic iterates Julia set Lemma Let f linear linearizable map f Math meromorphic maps multiplicity neighbourhood obtain open set orbit p.s.h. functions parabolic periodic points pluripolar pluripolar set plurisubharmonic polynomial polynomial-like map positive closed currents probability measure Proof properties Proposition prove rational surface real analytic sequence singular smooth space super-potential tangent Theorem theory topological degree totally invariant uniformly wandering domains zero