Appendix 1. Metric entropy

Suppose fi is invariant with respect to the transformation T, then the metric entropy Hu. D is defined rigorously as follows: let a = {A,} and /8 = {Bi}, (i = 1,2— n(a) or n(j0)). be partitions of the phase space I and let a(n) be defined as46           JJK Btmw, QNMfcMhMiNViiflNMMf NlMMlP

 

 

where Tl-*\a) is a partition of I into 7^(A,), Ti{A2\..., Tl(An{„y) and a v p is the partition of I into the sets A, HB, with independent i and j.

 

The entropy per unit step-lengtn of the partition a is defined as

 

and finally the metric entropy is defined as

The entropy of the partition a is defined as HM(«) where

 

 

where the supremum is taken over all hmte (or countable), measurable partitions ot the phase space.

Kolmogorov and Sinai [16] call a partition a generator if the diameters of the members of a(n) tend to

zero; as n -* » and they show that if a is a generator then i