Fractal string

Ordinary fractal strings.
An ordinary fractal string formula_1 is a bounded, open subset of the real number line. Any such subset can be written as an at-most-countable union of connected open intervals with associated lengths formula_2 written in non-increasing order. We allow formula_1 to consist of finitely many open intervals, in which case formula_4 consists of finitely many lengths. We refer to formula_4 as a "fractal string".
Example.
The middle third's Cantor set is constructed by removing the middle third from the unit interval formula_6, then removing the middle thirds of the subsequent intervals, "ad infinitum". The deleted intervals formula_7 have corresponding lengths formula_8. Inductively, we can show that there are formula_9 intervals corresponding to each length of formula_10. Thus, we say that the "multiplicity" of the length formula_11 is formula_12.
Heuristic.
The geometric information of the Cantor set in the example above is contained in the ordinary fractal string formula_4. From this information we can compute the box-counting dimension of the Cantor set. This notion of fractal dimension can be generalized to that of complex dimension, which will give us complete geometrical information regarding the local oscillations in the geometry of the Cantor set.
The geometric zeta function.
If formula_14 we say that formula_15 has a geometric realization in formula_16 formula_17, where the formula_18 are intervals in formula_19, of all the lengths formula_20, taken with multiplicity.
For each fractal string formula_4, we can associate to formula_4 a geometric zeta function formula_23 defined as the Dirichlet series formula_24. Poles of the geometric zeta function formula_25 are called complex dimensions of the fractal string formula_4. The general philosophy of the theory of complex dimensions for fractal strings is that complex dimensions describe the intrinsic oscillation in the geometry, spectra and dynamics of the fractal string formula_4.
The abscissa of convergence of formula_28 is defined as formula_29.
For a fractal string formula_4 with infinitely many nonzero lengths, the abscissa of convergence formula_31 coincides with the Minkowski dimension of the boundary of the string, formula_32. For our example, the boundary Cantor string is the Cantor set itself. So the abscissa of convergence of the geometric zeta function formula_33 is the Minkowski dimension of the Cantor set, which is formula_34.
Complex dimensions.
For a fractal string formula_4, composed of an infinite sequence of lengths, the "complex dimensions" of the fractal string are the poles of the analytic continuation of the geometric zeta function associated with the fractal string. (When the analytic continuation of a geometric zeta function is not defined to all of the complex plane, we take a subset of the complex plane called the "window", and look for the "visible" complex dimensions that exist within that window.)
Example.
Continuing with the example of the fractal string associated to the middle thirds Cantor set, we compute formula_36. We compute the abscissa of convergence to be the value of formula_37 satisfying formula_38, so that formula_39 is the Minkowski dimension of the Cantor set.
For complex formula_37, formula_41 has poles at the infinitely many solutions of formula_38, which, for this example, occur at formula_43, for all integers formula_44. This collection of points is called the set of complex dimensions of the middle thirds Cantor set.
Applications.
For fractal strings associated with sets like Cantor sets, formed from deleted intervals that are rational powers of a fundamental length, the complex dimensions appear in a regular, arithmetic progression parallel to the imaginary axis, and are called "lattice" fractal strings. Sets that do not have this property are called "non-lattice". There is a dichotomy in the theory of measures of such objects: an ordinary fractal string is Minkowski measurable if and only if it is non-lattice.
The existence of non-real complex dimensions with positive real part has been proposed to be the signature feature of fractal objects. Formally, Michel Lapidus and Machiel van Frankenhuijsen propose to define “fractality” as the presence of at least one nonreal complex dimension with positive real part. This new definition of fractality solves some old problems in fractal geometry. For example, everyone can agree that Cantor's devil's staircase is fractal, which it is with this new definition of fractality in terms of complex dimensions, but it is not in the sense of Mandelbrot.