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Valuations and Hyperbolicity in Dynamics

By Ward, Thomas

Book Id:WPLBN0000659769 Format Type:PDF eBook File Size:559.08 KB Reproduction Date:2005 Full Text

Ward, T. (n.d.). Valuations and Hyperbolicity in Dynamics. Retrieved from http://hawaiilibrary.net/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: Valuations And Hyperbolicity In Dynamics; A Hyperbolic Automorphism; S-integer dynamical systems; Definition and examples. The S-integer dynamical systems are a very simple collection of dynamical systems which are the pieces from which group automorphisms may be built up. Most of the material here is taken from [11]. An excellent modern treatment of Tate?s thesis and related material is the text of Ramakrishnan and Valenza, [57]. Let k be an A?field in the sense of Weil (that is, k is an algebraic extension of the rational field Q or of Fq(t) for some rational prime q), and let P(k) denote the set of places of k. A place w 2 P(k) is finite if w contains only non?archimedean valuations and is infinite otherwise (with one exception: for the case Fp(t) the place given by t?1 is regarded as being an infinite place despite giving rise to a non? archimedean valuation). Example 2.1. For the case k0 = Q or k0 = Fq(t), the places are defined as follows. The Rationals Q. The places of Q are in one?to?one correspondence with the set of rational primes {2, 3, 5, 7, . . . } together with one additional place 1 at infinity. The corresponding valuations are |r|1 = |r| (the usual archimedean valuation), and for each p, |r|p = p?ordp(r), where ordp(r) is the (signed) multiplicity with which the rational prime p divides the the rational r. The Function Field Fq(t). For Fq(t) there are no archimedean places. For each monic irreducible polynomial v(t) 2 Fq[t] there is a distinct place v, with corresponding valuation given by |f|v = q?ordv(f)?deg(v), where ordv(f) is the signed multiplicity with which v divides the rational function f. There is one additional place given by v(t) = t?1...