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By Lefschetz S.

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E. e. iff ~ and ~' are independent. e. as dm. 3)Definition: Hm(~) = The quantity -~_____re(A) log re(A) AE~ is called the entropy of the partition ~. We shall often omit the subscript m. (We define 0 log O = O). It should be noted that if ~ ~ ~' then H(~) = H(~'). Thus in dealing with entropy we need not distinguish between partitions and equivalence classes of partitions as we shall do in the following two sections. 4) Proposition: (a) 0 ~ H(~), with equality iff ~ is trivial; (b) If ~ has k atoms then H(~) ~ log k, with equality iff all 57 I atoms have measure ~.

Uj). "" + l One has "~1 = n j = l 6(y]) c v I and 33 TN I n v2 = ~----8(TNj=I yj2) E V2. Hence T-N V 2 N V 1 + ~ f o r a l l N ~ N O , and so the extended transformation is strongly mixing. (2) The weakly mixing case is analogous. 9) Proposition: T : X JX is top. transitive (resp. top. weakly mixing, resp. top strongly mixing) if its extension T : ~(X)--e~(X) top. weakly mixing, is top. transitive (resp. resp. top. strongly mixing). Proof: Assume that T : ~(X)--*~(X) is top. transitive. Let UI, U 2 be nonempty open in X.

16) Definition: (X,T) is said to be strictly ergodic if it is uniquely ergodic and minimal. 17) Definition: A point x E X is said to be nonwandering (with respect to T) if for every neighborhood U o2 x, there exists an n ~ 0 with U a T -n U ~ ~. The set of all nonwandering points is called the nonwanderin~ set and denoted by ~T(X). 18) Proposition: OT(X ) is a nonempty closed T-invariant subset of X. It contains all minimal sets, and in particular all periodic orbits. Note that in general the nonwandering set of the restriction of T to aT(X ) need not coincide with OT(X ).

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Continuous Transformations of Manifolds by Lefschetz S.

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