By Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D.S. Jones

In 1989-90 the Mathematical Sciences examine Institute carried out a application on Algebraic Topology and its functions. the most parts of focus have been homotopy concept, K-theory, and purposes to geometric topology, gauge conception, and moduli areas. Workshops have been performed in those 3 components. This quantity includes invited, expository articles at the subject matters studied in this application. They describe fresh advances and aspect to attainable new instructions. they need to turn out to be necessary references for researchers in Algebraic Topology and comparable fields, in addition to to graduate scholars.

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Lemma C and ~ and Proof. As Let T be an endomorphism of a probability space finite partitions. Then (a) H(T-kCIT-k~) (b) h(T,C) (c) h (T,C V . VT'a+lcI4 v.. ,VT-m+14) . 3 C Lemma Let X Proof. VT-m-n+Ic) ~ be a compact metric space, a finite Borel partition. VT-nC ! & Proposition. metric space, diam a n ~ E MT(X) ~ 0 T:X ~ X and that ~n is a continuous map of a compact is a sequence of partitions with Then h (T) = lim h~m Proof. 5 Prgposition. constant r Proof. Then Let expansiveness. vT-na Hence D hp(T) = lim h (T,~n) ~ E MT(X) , and dism Then diam an ~ 0 But a ~ c 9 using h (T,an) = h (T,~) by 4g Consider the case of where U i = Ix E EA : x o = i} h (a): h (a,U) for chapter 1.

6) I ~ (h~(Tn,~n) + ~ S n ~ ) is a partition SS n-I 9 Since is ~n k=O refines " T-kG k , for each T-invariant , T-kG one has (since H (T-kcl~ n ) < r bears the same relation to T-kc as G to C). 12, i h(T,C) + 2~ ~ -< PT(~) + ~ log nlci + . Now let n -. 13 Proposition. Let compact metric spaces, ~TI = T2, r - 0 . Ti: X - X , T2: Y ~ Y w : ~ continuous and onto satisfying - X2 be continuous maps on Then PT2 (~) -< PTI(~O~) for ~ E c(~) Proof. ,~-Z~) = y(~,~) - 0 as aiam ~ ~ 0 - 0 we get PT2 (@) -< PTl(e~ 90 Hence, letting dism 56 Variational Principle C.

Constant Because (~o~)~ . derivative is m~ique up to M-equivalence, ~ is ergodlc this gives c . l=~'(Z A) = I c ~ = c and ~' = ~ . [3 f f~o equivalent to some 28 D. 1. e. the C i k disjoint and X = U C i) , o~e defines the entropy i=l is a are pairwise k H(C)= ~. 17 ~ . H(CV~) 0 i- Applying this to [0,i] as 29 &i=~(Ci) and ~(Ci N D i ) ~(Ci ) xi= ~ ~(Ci)r i we get (Ci n DJ ) ~(Ci ) )- | , lira am .

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