First, we show that each q determines a unique value of tau(q). |

Next, tau(q) -> infinity as q -> -infinity, and
tau(q) -> -infinity as q -> infinity. |

Tau is a decreasing function of q and is concave up. |

Corresponding to each q, say alpha is the negative of the slope of the tangent to the graph of tau(q).
This tangent line intersects the y-axis at a value called f(alpha). |

The range of alpha values is min{log(p_{i})/log(r_{i})}
= alpha_{min} <= alpha <= alpha_{max} = max{log(p_{i})/log(r_{i})}. |

The graph of tau(q) has oblique asymptotes, with slopes -alpha_{max}
as q -> -infinity and -alpha_{min} as q -> infinity. |

If all the log(p_{i})/log(r_{i}) are distinct, then f(alpha_{min}) =
f(alpha_{max}) = 0. |

If not all the log(p_{i})/log(r_{i}) are equal, then the graph of f(alpha) is
concave down. |

The alpha values and local dimensions have the same range, and f(alpha) is the dimension of the
set with local dimension alpha. That is, f(alpha) = dim(E_{alpha}). |

The maximum value of the f(alpha) curve (which occurs at q = 0) is the dimension of the attractor
of the IFS. |

The f(alpha) curve intersects the line y = x at a single point (corresponding to q = 1) giving a
value called the dimension of the measure. |