Entropy of n-Fold Compositions of the Tent Map
Entropy of n-Fold Compositions of the Tent Map
Define a tent map by
f:[0,1]
+
f(x)=
α β |
α 1-β |
0,x=1 |
where , ; has a maximum at .
0≤β<1
α>0
f
(β,α)
Let be the function composed with itself times and let be the number of intervals over which is monotone; for the tent maps considered here, is twice the number of maxima of .
n
f
f
n
C
n
n
f
C
n
n
f
Define the entropy to be .
S(f)=
lim
n∞
log
2
C
n
n
For the tent map with maximum at , the entropy of is 1, as the number of maxima is =.
(1/2,1)
C
n
n
2