World Library  


Add to Book Shelf
Flag as Inappropriate
Email this Book

Valuations and Hyperbolicity in Dynamics

By Ward, Thomas

Click here to view

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

Title: Valuations and Hyperbolicity in Dynamics  
Author: Ward, Thomas
Volume:
Language: English
Subject: Science., Mathematics, Logic
Collections:
Historic
Publication Date:
Publisher:

Citation

APA MLA Chicago

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

Table of Contents
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. S-integer dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Definition and examples 3 2.2. Background on adeles 7 2.3. Adelic covering space 8 2.4. Topological entropy 10 2.5. Dynamical properties 12 2.6. Periodic points 13 2.7. Growth rates 15 2.8. Typical group automorphisms 19 3. Bernoullicity and recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1. Automorphisms of solenoids 25 3.2. Exponential recurrence 27 3.3. Commuting automorphisms 27 4. Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1. Background from algebra 30 4.2. Order of mixing ? connected case 32 4.3. Order of mixing ? disconnected case 33 4.4. Typical actions 40 5. Subdynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.1. Examples 44 5.2. Adelic amoebas 47 6. Some directions for future research . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1. Typical group automorphisms 49 6.2. Periodic points 49 6.3. Mixing problem 50 6.4. Entropy 50 6.5. Entropy and Deligne periods 50 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

 

Click To View

Additional Books


  • Resurrectio Divi Quirini Francisci Bacon... (by )
  • New Methods in Analytical Chemistry (by )
  • Contents 
  • Die Irrationalen Quadratwurzeln Bei Arch... (by )
  • Introduction to Mechanics and Symmetry Q (by )
  • Vorlesungen über Mathematische Physik (by )
  • Current Textbooks in Algorithmic Algebra... 
  • The Point of View (by )
  • The Pension Beaurepas (by )
  • The Patagonia (by )
  • The Papers (by )
  • The Papers (by )
Scroll Left
Scroll Right

 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library on the Kindle are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.