Taking limits as the side length of the regions go to zero, the coarse Holder exponent
can be refined to the local Holder exponent (or roughness) at |

d_{loc}(x,y) = lim_{n -> infinity} Log(Prob(i_{1}...i_{n}))/Log(2^{-n}) |

where Prob(i_{1}...i_{n}) is the probability pr(i_{1})* ... *pr(i_{n}), if _{1}...i_{n} |

The value for a square of finite length address is called the coarse Holder exponent.
So the local Holder exponent of a point |

Now define |

E_{alpha} = {(x, y): d_{loc}(x, y) = alpha}, |

the collection of all points of the fractal having local Holder exponent alpha. |

As alpha takes on all values of the local Holder exponent, we decompose the fractal
into these sets E_{alpha}. |

Here are examples, E_{alpha} (alpha = column height) for the lowest value of alpha (on the left), two
intermediate values, and the highest value. |

Click
here for an animation scanning through all the values of
alpha, from lowest to highest, resolved to boxes have side length ^{4}. |

Because each local Holder exponent alpha is the exponent for a power law, a multifractal is a process exhibiting scaling for a range of different power laws. |

The multifractal structure is revealed by plotting
_{alpha}) |

(In general, a dimension more subtle than the box-counting dimension must be used. We ignore this complication here.) |

This graph is called the f(alpha) curve. |

Here is the f(alpha) curve for the example with
_{1} = 0.2,_{2} = p_{3} = 0.25,_{4} = 0.3. |

At least in this example, sets E_{alpha} for the lowest and highest values of alpha
reduce to points in the limit, hence have dimension |

This result is derived under more general conditions in a later section. |

Return to Multifractals.