WOLFRAM|DEMONSTRATIONS PROJECT

Entropy of n-Fold Compositions of the Tent Map

maximum at n = 1

number of compositions n

1

Define a tent map

f:[0,1]

+



by

f(x)=

α

β

x,0≤x≤β

α

1-β

(1-x),β≤x≤1

0,x=1

where

0≤β<1

,

α>0

;

f

has a maximum at

(β,α)

.

Let

n

f

be the function

f

composed with itself

n

times and let

C

n

be the number of intervals over which

n

f

is monotone; for the tent maps considered here,

C

n

is twice the number of maxima of

n

f

.

Define the entropy to be

S(f)=

lim

n∞

log

2

C

n

n

.

For the tent map with maximum at

(1/2,1)

, the entropy of is 1, as the number of maxima is

C

n

=

n

2

.