An ordinary fractal string
is a bounded, open subset of the
real number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
. Such a subset can be written as an at-most-
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
union of connected
open intervals with associated lengths
written in non-increasing order; we also refer to
as a fractal string. For example,
is a fractal string corresponding to the
Cantor set. A fractal string is the analogue of a one-dimensional "fractal drum," and typically the set
has a boundary
which corresponds to a fractal such as the Cantor set. The heuristic idea of a fractal string is to study a (one-dimensional) fractal using the "space around the fractal." It turns out that the sequence of lengths
of the set itself is "intrinsic," in the sense that the fractal string
itself (independent of a specific geometric realization of these lengths as corresponding to a choice of set
) contains information about the fractal to which it corresponds.
For each fractal string
, we can associate to
a geometric zeta function
: the
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
. Informally, the geometric zeta function carries geometric information about the underlying fractal, particularly in the location of its
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
and the
residues of the zeta function at these poles. These poles of (the
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
of) the geometric zeta function
are then called complex dimensions of the fractal string
, and these complex dimensions appear in formulae which describe the geometry of the fractal.
For fractal strings associated with sets like Cantor sets, formed from deleted intervals that are
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
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
, 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
is defined to be a local positive or complex measure on
such that
for some
, where the positive measure
is the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
measure associated to
. These generalized fractal strings allow for lengths to be given non-integer multiplicities (among other possibilities), and each ordinary fractal string can be associated with a measure that makes it into a generalized fractal string.
Ordinary fractal strings
An ordinary
fractal string
is a bounded, open subset of the real number line. Any such subset can be written as an at-most-
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
union of connected
open intervals with associated lengths
written in non-increasing order. We allow
to consist of finitely many open intervals, in which case
consists of finitely many lengths. We refer to
as a ''fractal string''.
Example
The
middle third's Cantor set is constructed by removing the middle third from the unit interval
, then removing the middle thirds of the subsequent intervals, ''ad infinitum''. The deleted intervals
have corresponding lengths
. Inductively, we can show that there are
intervals corresponding to each length of
. Thus, we say that the ''multiplicity'' of the length
is
. The fractal string of the Cantor set is called the ''Cantor string''.
Heuristic
The geometric information of the Cantor set in the example above is contained in the ordinary fractal string
. From this information, we can compute the
box-counting dimension of the Cantor set. This notion of
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
can be generalized to that of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
, which may be used to deduce geometrical information regarding the local oscillations in the geometry of the fractal. For example, the complex dimensions of a fractal string (such as the Cantor string) may be used to write an explicit tube formula for the volume of an
-neighborhood of the fractal string, and the presence of non-real complex dimensions corresponds to oscillatory terms in this expansion.
The geometric zeta function
If
we say that
has a geometric realization in
, where the
are intervals in
, of all the lengths
, taken with multiplicity.
For each fractal string
, we can associate to
a geometric zeta function
defined as the Dirichlet series
.
Poles of the geometric zeta function
are called complex dimensions of the fractal string
. 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
.
The
abscissa of convergence In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
: \sum_^\infty a_n e^,
where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends t ...
of
is defined as
.
For a fractal string
with infinitely many nonzero lengths, the abscissa of convergence
coincides with the
Minkowski dimension Minkowski, Mińkowski or Minkovski (Slavic feminine: Minkowska, Mińkowska or Minkovskaya; plural: Minkowscy, Mińkowscy; he, מינקובסקי, russian: Минковский) is a surname of Polish origin. It may refer to:
* Minkowski or Mińko ...
of the boundary of the string,
.
For our example, the boundary Cantor string is the Cantor set itself. So the abscissa of convergence of the geometric zeta function
is the Minkowski dimension of the Cantor set, which is
.
Complex dimensions
For a fractal string
, 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 . We compute the abscissa of convergence In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
: \sum_^\infty a_n e^,
where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends t ...
to be the value of satisfying , so that is the Minkowski dimension Minkowski, Mińkowski or Minkovski (Slavic feminine: Minkowska, Mińkowska or Minkovskaya; plural: Minkowscy, Mińkowscy; he, מינקובסקי, russian: Минковский) is a surname of Polish origin. It may refer to:
* Minkowski or Mińko ...
of the Cantor set. For complex , has poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
at the infinitely many solutions of , which, for this example, occur at , for all integers . This collection of points is called the set of complex dimensions of the middle thirds Cantor set.
Applications
Ordinary and generalized fractal strings may be used to study the geometry of a (one-dimensional) fractal, as well as to relate the geometry of the object to its spectrum. For example, the geometric zeta function associated to a fractal string may be used to write an explicit tube formula for the volume of a neighborhood of the fractal. Regarding the connection between geometry and spectra, the ''spectral zeta function'' of a fractal string, which is the geometric zeta function times the Riemann zeta function, may be used to write explicit formulae which describe spectral counting functions.
The framework of fractal strings also serves to unify aspects of fractal and arithmetic geometry. For example, a general explicit formula for counting the (reciprocal) lengths of a fractal string may be used to prove Riemann's explicit formula when using a suitable generalized fractal string which is supported on the prime powers with multiplicities of each given by the logarithm of the prime base of the power.
For fractal strings associated with sets like Cantor sets, formed from deleted intervals that are rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
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 by Michel Lapidus and Machiel van Frankenhuijsen to be the signature feature of fractal objects.[M. L. Lapidus, M. van Frankenhuijsen]
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, they 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, according to the proposed definition of fractality in the sense of Mandelbrot, Cantor's devil's staircase not fractal because its Hausdorff and topological dimensions coincide. However, the Cantor staircase function possesses many features which ought to be considered fractal such as self-similarity, and in this new sense of fractality the Cantor staircase function is considered fractal since it has non-real complex dimensions.
Generalized fractal strings
A generalized fractal string is defined to be a local positive or local complex measure on such that for some , where the positive measure is the total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
measure associated to . A generalized fractal string allows for a fractal string to have a given set of lengths with non-integer multiplicities, or for a fractal string to have a continuum of lengths instead of discrete. By convention, a generalized fractal string is supported on reciprocal lengths as opposed to an ordinary fractal string which is a multiset of (decreasing or non-increasing) lengths. In light of this, the condition that the measure has "no mass near zero," or more precisely that there exists a positive number such that the interval has measure zero with respect to , may be seen as an analogue of the boundedness of the ordinary fractal string.
For example, if is an ordinary fractal string with multiplicities , then the measure associated to (where refers to the Dirac delta measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
...
concentrated at the point ) is an example of a generalized fractal string. Note that the delta functions are supported on the singleton sets corresponding to the reciprocals of the lengths of the ordinary fractal string . If the multiplicities are not positive integers, then is a generalized fractal string which cannot be realized as an ordinary fractal string. A concrete example of such a generalized fractal string would be the ''generalized Cantor string'' for .
If is a generalized fractal string, then its ''dimension'' is defined as its ''counting function'' as
and its ''geometric zeta function'' (its Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used ...
) as
(Note that the counting function is normalized at jump discontinuities to be half of the value at any singletons which have nonzero measure.)
References
{{fractals, state=collapsed
Fractals
Mathematical structures
Iterated function system fractals
Sets of real numbers