# 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