By K. Tomita (auth.), Professor Dr. Yoshiki Kuramoto (eds.)

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Extra info for Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan September 12–15, 1983

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The map F on J=[O,l] is defined by the conditioRs thlt it is coHtinuous, that it is linear on each subinterval I i =[c i ,c i _ 1 ] and that F(c i )=c i _ 1 (i~l) and F(CO)=O. Then, 21 F (Ii) =I i _ l (i2:,l) and F(IO)=J. } plays the role of the free orbit basis. It then follows immediat§ly that n+l with b n =II n I=c n -c n+l 00(z)= 1- L b z , PO= log 2, n n+l 0l(z)= 1- L bnz , Pl=O. Hence the invariant density exists and is a probability density or not according to -Oi(l-O)= L (n+l)b n <00 or = 00.

In this case, the construction shows that Do (l/S)=O and so P O=log S is the topological entropy. Since F' is constant, n+1 D1 (Z)= 1- L an(z/S) , P 1 =0, 0 1 (1)=0, Di(l) finite. This implies that F shows observaBle chaosn+ 1n this case, the invariant density is u*(t)= const. l[t F (1-0)]/ • C. Unimodal Linear Maps: Ft= (a+b-1+t)/b (l-t)/a on I O=[O,l-a] on I 1 =[1-a,1] where OO and a+b~l. The case a+b=l is the tent map. Math. 2(1979). It is shown, in my terminology, that a free orbit basis exists and so F shows observable chaos except for the following two cases (p~3): (a) ab P - 1 <1, a(a+b)bP-2~1, a(1+b+ ••• +b P - 2 )<1

Bowen, Publ. Math. IHES 50, 11 (1979). M. Hidom, D. J. Shenker, J. Stat. Phys. to appear (1983). 4. D. Ruelle, Invert. Math. 34, 231 (1976). 5. R. Bowen, Equi Ubrium States and the Ergodic Theory' of Anosov Diffeomorphism, Springer (Berlin) Lecture Notes in Math. 470 (1975). 6. B. Y. Academy-of Sciences 357, 249 (1980). 7. For an excellent review article on theorems of comolexlmappings see H. Brolin, Arkiv For Matematik 6, 103 (1965). 8. D. Ruelle, TheY'TTIodynamic FOY'TTIalism, Addison-l-Jesley:Reading (1978).

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