# Download e-book for kindle: Algebraic Methods in the Global Theory of Complex Spaces by Constantin Banica, Octavian Stanasila

By Constantin Banica, Octavian Stanasila

ISBN-10: 0471018090

ISBN-13: 9780471018094

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**Sample text**

2) Show that the action of Q (IIJ)) on the boundary circle aIIJ) carries three specified points into three other specified points if and only if they have the same cyclic order. Problem 2-e. Classifying automorphisms of IIJ). Show that an automorphism of lHI or IIJ) is hyperbolic (Problem I-d) if and only if it carries some Poincare geodesic into itself without fixed points. Problem 2-f. Infinite strip, cylinder, and annulus. Define the infinite strip Bee of height 7f to be the set of all z == x + iy with Iy I < 7f / 2 .

For any compact K C X and any f E Map(X, Y), let NK,U(f) = {g E Map(X, Y) ; (f(x), g(x)) E U for all x E K }. Given K and U, it is not difficult to construct an E > 0 so that every pair (f(x), y) with x E K and dist(f(x), y) < E belongs to this set U, and it then follows that NK,E(f) C NK,U(f). ) = NK,u(E)(f). Thus, if we take {NK,u(f)} as our collection of "basic neighborhoods," then we obtain the same topology, without mentioning any particular choice of metric. Now suppose that X is a-compact; that is, suppose that X is a countable union of compact subsets.

D The case of an indifferent periodic point is much more difficult. ) One particularly important case is the following. Definition. A periodic point Zo == jon(zo) is called parabolic if the multiplier A at Zo is equal to +1, yet fon is not the identity map, or more generally if A is a root of unity, yet no iterate of f is the identity map. As an example, the rational map j(z) == z/(z-l) has two fixed points, both with multiplier equal to -1. However, these do not count as parabolic points since f 0 j(z) is identically equal to z.

### Algebraic Methods in the Global Theory of Complex Spaces by Constantin Banica, Octavian Stanasila

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