Mandelbrot's Binomial Measure Multifractal

m

0

0.3

iterations

7

This Demonstration illustrates the simplest multifractal: Mandelbrot's binomial measure on the interval

[0,1]

. The iteration begins with a uniform distribution

μ

0

[0,1]

m

0

(with

0<

m

0

<1

), subdivides it into a distribution with

μ

1

[0,1/2]

m

0

and

μ

2

[1/2,1]

m

1

1-

m

0

, further subdivides it into

μ

2

[0,1/4]

m

0

m

0

,

μ

2

[1/4,1/2]

m

0

m

1

,

μ

2

[1/2,3/4]

m

1

m

0

, and

μ

2

[3/4,1]

m

1

m

1

, and so on. Additional iteration of this procedure gives a multiplicative cascade that generates an infinite sequence of measures; the limit of the measures is the binomial measure.