An ordinary fractal string $\backslash omega$ is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable set, countable union of connected open intervals with associated lengths $\backslash mathcal=\backslash $ written in non-increasing order; we also refer to $\backslash mathcal$ as a fractal string. For example, $\backslash mathcal=\backslash left\backslash $ is a fractal string corresponding to the Cantor set.
For each fractal string $\backslash mathcal$, we can associate to $\backslash mathcal$ a geometric zeta function $\backslash zeta\_$: the Dirichlet series $\backslash zeta\_\; (s)=\backslash sum\_\; \backslash ell\_j^$. Poles of (the analytic continuation of) the geometric zeta function $\backslash zeta\_\; (s)$ are then called complex dimensions of the fractal string $\backslash mathcal$. For fractal strings associated with sets like Cantor sets, formed from deleted intervals that are rational number, rational powers of a fundamental length, the complex dimensions appear in an arithmetic progression parallel to the imaginary axis, and are called lattice fractal strings. (For example the complex dimensions of the Cantor set are $s=\backslash frac$, which are an arithmetic progression in the direction of the imaginary axis.) Otherwise, they are called non-lattice. In fact, an ordinary fractal string is Minkowski measurable if and only if it is non-lattice.
A generalized fractal string $\backslash eta$ is defined to be a local positive or complex measure on $(0,\; +\backslash infty)$ such that $,\; \backslash eta,\; (0,\; x\_0)\; =\; 0$ for some $x\_0\; >\; 0$, where the positive measure $,\; \backslash eta,$ is the variation of $\backslash eta$. Each ordinary fractal string can be associated with a measure that makes it into a generalized fractal string.

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

Monographs in Mathematics, Springer, New York, Second revised and enlarged edition, 2012. 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 function, 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.

Ordinary fractal strings

An ordinary fractal string $\backslash omega$ is a bounded, open subset of the real number line. Any such subset can be written as an at-most-countable set, countable union of connected open intervals with associated lengths $\backslash mathcal=\backslash $ written in non-increasing order. We allow $\backslash omega$ to consist of finitely many open intervals, in which case $\backslash mathcal$ consists of finitely many lengths. We refer to $\backslash mathcal$ as a ''fractal string''.Example

The Cantor ternary set, middle third's Cantor set is constructed by removing the middle third from the unit interval $(0,1)$, then removing the middle thirds of the subsequent intervals, ''ad infinitum''. The deleted intervals $\backslash Omega=\backslash left\backslash $ have corresponding lengths $\backslash mathcal=\backslash left\backslash $. Inductively, we can show that there are $2^$ intervals corresponding to each length of $3^$. Thus, we say that the ''multiplicity'' of the length $3^$ is $2^$.Heuristic

The geometric information of the Cantor set in the example above is contained in the ordinary fractal string $\backslash mathcal$. 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 $\backslash sum\_\; <\; \backslash infty,$ we say that $\backslash Omega$ has a geometric realization in $\backslash mathbb,$ $\backslash Omega=\backslash bigcup\_^\backslash infty\; I\_i$, where the $I\_i$ are intervals in $\backslash mathbb$, of all the lengths $\backslash \_$, taken with multiplicity. For each fractal string $\backslash mathcal$, we can associate to $\backslash mathcal$ a geometric zeta function $\backslash zeta\_$ defined as the Dirichlet series $\backslash zeta\_\; (s)=\backslash sum\_\; \backslash ell\_j^$. Poles of the geometric zeta function $\backslash zeta\_\; (s)$ are called complex dimensions of the fractal string $\backslash mathcal$. 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 $\backslash mathcal$. The abscissa of convergence of $\backslash zeta\_(s)$ is defined as $\backslash sigma=\backslash inf\; \backslash left\backslash $. For a fractal string $\backslash mathcal$ with infinitely many nonzero lengths, the abscissa of convergence $\backslash sigma$ coincides with the Minkowski dimension of the boundary of the string, $\backslash partial\; \backslash Omega$. For our example, the boundary Cantor string is the Cantor set itself. So the abscissa of convergence of the geometric zeta function $\backslash zeta\_(s)$ is the Minkowski dimension of the Cantor set, which is $\backslash frac$. The concept of a geometric zeta function can be generalized to higher dimensions as the ''distance zeta function''.Complex dimensions

For a fractal string $\backslash mathcal$, 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 $\backslash zeta\_(s)=\backslash sum\_^\backslash infty\; \backslash frac\; =\backslash frac\; =\; \backslash frac$. We compute the abscissa of convergence to be the value of $s$ satisfying $3^s=2$, so that $s=\backslash log\_3\; 2=\backslash frac$ is the Minkowski dimension of the Cantor set. For complex $s$, $\backslash zeta\_(s)$ has pole (complex analysis), poles at the infinitely many solutions of $3^s=2$, which, for this example, occur at $s=\backslash frac$, for all integers $k$. 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 number, 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.M. L. Lapidus, M. van FrankenhuijsenFractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

Monographs in Mathematics, Springer, New York, Second revised and enlarged edition, 2012. 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 function, 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.

Generalized fractal strings

A generalized fractal string $\backslash eta$ is defined to be a local positive or complex measure on $(0,\; +\backslash infty)$ such that $,\; \backslash eta,\; (0,\; x\_0)\; =\; 0$ for some $x\_0\; >\; 0$, where the positive measure $,\; \backslash eta,$ is the variation of $\backslash eta$. For example, if $\backslash mathcal\; =\; \backslash \_^$ is an ordinary fractal string with multiplicities $w\_j$, then the measure $\backslash eta\_\; :=\; \backslash sum\_^\; w\_j\backslash delta\_$ associated to $\backslash mathcal$ (where $\backslash delta\_$ refers to the Dirac delta measure concentrated at the point $x$) is an example of a generalized fractal string. Even more concretely, one may consider, for example, the ''generalized Cantor string'' $\backslash eta\_\; :=\; \backslash sum\_^\; b^j\; \backslash delta\_$ for $1\; <\; b\; <\; a$. If $\backslash eta$ is a generalized fractal string, then its ''dimension'' is defined as $$D\_\; :=\; \backslash inf(\backslash sigma\backslash in\backslash mathbb:\; \backslash int\_0^\; x^,\; \backslash eta,\; (dx)\; <\; \backslash infty),$$its ''counting function'' as $$N\_(x)\; :=\; \backslash int\_0^x\; \backslash eta(dx)\; =\; \backslash eta(0,\; x)$$and its ''geometric zeta function'' (its Mellin transform) as $$\backslash zeta\_(s)\; :=\; \backslash int\_0^\; x^\backslash eta(dx).$$References

{{reflist Fractals Mathematical structures Iterated function system fractals Sets of real numbers