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Appendix B : Lawton's Proof of Theorem 9. 6

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Book Id: WPLBN0000654340
Format Type: PDF eBook
File Size: 166.82 KB
Reproduction Date: 2005
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Title: Appendix B : Lawton's Proof of Theorem 9. 6  
Language: English
Subject: Science., Mathematics, Logic
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Appendix B : Lawton's Proof of Theorem 9. 6. (n.d.). Appendix B : Lawton's Proof of Theorem 9. 6. Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: In this appendix we present Lawton's proof of Theorem 9.4. The proof follows Lawton [17] closely, and uses standard results on topological entropy (from Chapters 6 and 7). Let G be a compact abelian group, with T an automorphism of G. If H G is a T{invariant closed subgroup, then denote by TH the automorphism of H obtained by restricting T to H. Definition B.1. The dual group of G, bG, is said to be nitely generated under bT if there are elements 1; n in bG such that f bTk( i) j i = 1; n; k 2 Zg generates bG. Notice that the dual group bG is discrete (since G is compact), so that f bTk( i) j i = 1; n; k 2 Zg generates if and only if it separates points of G. Thus, to show that bG is nitely generated under bT, it is enough to find 1; n in bG with the property that for every g 2 Gnf0g there exists a k 2 Z and an i 2 f1; ng for which ( bTk i)(g) = i(Tkg) 6= 0...


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