An improved multifractal formalism and self a ne measures

It has been recognized that most fractals K in nature are actually composed of an in nite set of interwoven subfractals This structure becomes apparent when a particular measure of total mass supported by K is considered To every singularity exponent belongs a set C of all points of K for which the measure of the balls with radius roughly scales as for These sets are usually fractals giving the name multifractal The complexity of the geometry of C is measured by the spectrum f which can be thought of as representing the box dimension of C However the singularities of may also be measured through the generalized dimensions dq Spectrum and generalized dimensions are very helpful when comparing multifractals appearing in nature with analytically treatable measures One central fact of the multi fractal formalism is the close relation between dq and f the convex q q dq is the Legendre transform of the concave f This allows to reduce the somewhat tedious if not impossible computation of f to the simpler one of dq Though widely used the various de nitions of f and dq di er only slightly A math ematically precise de nition as well as the important relation q sup f q can be found in F But unfortunately this concept turns out to be unsatisfactory for two reasons First of all Falconers f is de ned through a double limes which usually does not exist for great Secondly the generalized dimensions usually take the irrelevant value dq for negative q More concretely for as simple multifractals as the middle third Cantor measure half of the spectrum is lost and proposition in F concerning the Legendre relation cannot be applied The concept we propose meets the two mentioned problems by a simple improvement Instead of B Q lk lk taken from a grid G of size we use a kind of parallel body B Q lk lk This renders a measurement of the singularities of which depends more regularly on