42
2 Fundamental Concepts
Fig. 2.14 (a) The four elements of the partition resulting from the intersection of partitions α and
T α. (b) The eight elements of the partition resulting from the intersection of partitions α, T α, and
T 2 α. Each element is represented by a different pattern
the fraction of the area of the unit square that it occupies. Then
2
i=1 p
(0)
i
= 1.
Thus, the measure of an element is the area that it occupies. From Fig. 2.13, we see
that T n α will contain 2 n elements, T n α = (A
(n)
1 , . . . , A
(n)
2 n ). Let us next introduce
the partition α ∨ T α, which consists of elements A
(0)
i ∩ A
(1)
j (i, j = 1, 2), where
∩ denotes the intersection of the elements A
(0)
i and A
(1)
j . The partition α ∨ T α is
shown in Fig. 2.14a. Similarly. the elements of the partition α ∨T α ∨T 2 α are shown
in Fig. 2.14b.
The KS metric entropy can now be defined as
h KS (T ) = sup h(α, T ) = sup lim
n→∞
h(α ∨ T α ∨ . . . ∨ T n−1 α)
n
,
(2.85)
where
h(α) = −
i
p i ln(p i )
(2.86)
and the sum is taken over all elements of partition α. The maximum value of the
entropy occurs when the elements of a partition all have equal area. If we assume
that our partitions do have equal area, then it is easy to see that
h(α ∨ T α ∨ . . . ∨ T
n−1
α) = −
2 n
i=1
1
2
n
ln
1
2
n
= n ln(2).
(2.87)
Thus, for the baker’s map,
h KS (t) = ln(2).
(2.88)
This analysis can be extended to Bernoulli shifts with an alphabet with k “letters.” In
that case, the KS metric entropy is ln(k). Therefore, the baker’s map and Bernoulli
shifts in general are K-flows. The dynamics causes contraction in the q direction