文档介绍：APPENDIX B: LAWTON’S PROOF OF THEOREM
In this appendix we present Lawton’s proof of Theorem . The proof follows
Lawton [17] closely, and uses standard results on topological entropy (from Chapters
6 and 7).
Let G be pact 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.
Definiton . The dual group of G, G, is said to be finitely generated under
k
T if there are elements χ1, . . . , χn in G suchb that {T (χi) | i = 1, . . . , n; k ∈ Z}
generatesb G. b b
b k
Notice that the dual group G is discrete (since G pact), so that {T (χi) |
i = 1, . . . , n; k ∈ Z} generates ifb and only if it separates points of G. Thus, tob show
that G is finitely generated under T , it is enough to find χ1, . . . , χn in G with the
propertyb that for every g ∈ G\{0}bthere exists a k ∈ Z and an i ∈{1,b . . . , n} for
k k
which (T χi)(g) = χi(T g) =6 0.
b
Lemma . If T is an expansive automorphism of G, then G is finitely gener-
ated under T . b
b
Proof. Let U be an expansive neighbourhood of the identity in G, so that
() T k(U) = {e}.
\
k∈Z
Since characters separate the points of G, for every g ∈ G\U there is a character
χg ∈ G with χg(g) =6 0. By continuity, we may find, for each g ∈ G\U, an open
neighbourhoodb Ug of g with the property that χg(