Fractal string

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Seven iterations of the construction of the Cantor ternary set, an example of a fractal string.

An ordinary fractal string ${\displaystyle \omega }$ is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths ${\displaystyle {\mathcal {L}}=\{\ell _{1},\ell _{2},\ldots \}}$ written in non-increasing order; we also refer to ${\displaystyle {\mathcal {L}}}$ as a fractal string. For example, ${\displaystyle {\mathcal {L}}=\left\{{\frac {1}{3}},{\frac {1}{9}},{\frac {1}{9}},\ldots \right\}}$ is a fractal string corresponding to the Cantor set.

For each fractal string ${\displaystyle {\mathcal {L}}}$, we can associate to ${\displaystyle {\mathcal {L}}}$ a geometric zeta function ${\displaystyle \zeta _{\mathcal {L}}}$: the Dirichlet series ${\displaystyle \zeta _{\mathcal {L}}(s)=\sum _{j\in \mathbb {J} }\ell _{j}^{s}}$. Poles of (the analytic continuation of) the geometric zeta function ${\displaystyle \zeta _{\mathcal {L}}(s)}$ are then called complex dimensions of the fractal string ${\displaystyle {\mathcal {L}}}$. 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 an arithmetic progression parallel to the imaginary axis, and are called lattice fractal strings. (For example the complex dimensions of the Cantor set are ${\displaystyle s={\frac {\log 2+2\pi ik}{\log 3}}}$, 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 ${\displaystyle \eta }$ is defined to be a local positive or complex measure on ${\displaystyle (0,+\infty )}$ such that ${\displaystyle |\eta |(0,x_{0})=0}$ for some ${\displaystyle x_{0}>0}$, where the positive measure ${\displaystyle |\eta |}$ is the variation of ${\displaystyle \eta }$.^{[clarification needed]} 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 ${\displaystyle \omega }$ 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 ${\displaystyle {\mathcal {L}}=\{\ell _{1},\ell _{2},\ldots \}}$ written in non-increasing order. We allow ${\displaystyle \omega }$ to consist of finitely many open intervals, in which case ${\displaystyle {\mathcal {L}}}$ consists of finitely many lengths. We refer to ${\displaystyle {\mathcal {L}}}$ as a fractal string.

Example[]

The middle third's Cantor set is constructed by removing the middle third from the unit interval ${\displaystyle (0,1)}$, then removing the middle thirds of the subsequent intervals, ad infinitum. The deleted intervals ${\displaystyle \Omega =\left\{\left({\frac {1}{3}},{\frac {2}{3}}\right),\left({\frac {1}{9}},{\frac {2}{9}}\right),\left({\frac {7}{9}},{\frac {8}{9}}\right),\ldots \right\}}$ have corresponding lengths ${\displaystyle {\mathcal {L}}=\left\{{\frac {1}{3}},{\frac {1}{9}},{\frac {1}{9}},\ldots \right\}}$. Inductively, we can show that there are ${\displaystyle 2^{n-1}}$ intervals corresponding to each length of ${\displaystyle 3^{-n}}$. Thus, we say that the multiplicity of the length ${\displaystyle 3^{-n}}$ is ${\displaystyle 2^{n-1}}$.

Heuristic[]

The geometric information of the Cantor set in the example above is contained in the ordinary fractal string ${\displaystyle {\mathcal {L}}}$. 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.^{[clarification needed]}

The geometric zeta function[]

If ${\displaystyle \sum _{j\in \mathbb {J} }{\ell _{j}}<\infty ,}$ we say that ${\displaystyle \Omega }$ has a geometric realization in ${\displaystyle \mathbb {R} ,}$ ${\displaystyle \Omega =\bigcup _{i=1}^{\infty }I_{i}}$, where the ${\displaystyle I_{i}}$ are intervals in ${\displaystyle \mathbb {R} }$, of all the lengths ${\displaystyle \{\ell _{j}\}_{j\in \mathbb {J} }}$, taken with multiplicity.^{[citation needed]}

For each fractal string ${\displaystyle {\mathcal {L}}}$, we can associate to ${\displaystyle {\mathcal {L}}}$ a geometric zeta function ${\displaystyle \zeta _{\mathcal {L}}}$ defined as the Dirichlet series ${\displaystyle \zeta _{\mathcal {L}}(s)=\sum _{j\in \mathbb {J} }\ell _{j}^{s}}$.^[1] Poles of the geometric zeta function ${\displaystyle \zeta _{\mathcal {L}}(s)}$ are called complex dimensions of the fractal string ${\displaystyle {\mathcal {L}}}$. 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^{[weasel words]} of the fractal string ${\displaystyle {\mathcal {L}}}$.^{[citation needed]}

The abscissa of convergence of ${\displaystyle \zeta _{\mathcal {L}}(s)}$ is defined as ${\displaystyle \sigma =\inf \left\{\alpha \in \mathbb {R} :\sum _{j=1}^{\infty }\ell _{j}^{\alpha }<\infty \right\}}$.^[1]

For a fractal string ${\displaystyle {\mathcal {L}}}$ with infinitely many nonzero lengths, the abscissa of convergence ${\displaystyle \sigma }$ coincides with the Minkowski dimension of the boundary of the string, ${\displaystyle \partial \Omega }$.^[1] For our example, the boundary Cantor string is the Cantor set itself. So the abscissa of convergence of the geometric zeta function ${\displaystyle \zeta _{\mathcal {L}}(s)}$ is the Minkowski dimension of the Cantor set, which is ${\displaystyle {\frac {\log 2}{\log 3}}}$.^{[citation needed]}

The concept of a geometric zeta function can be generalized to higher dimensions as the distance zeta function.^{[2][how?][relevant?]}

Complex dimensions[]

For a fractal string ${\displaystyle {\mathcal {L}}}$, 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.^[3])^[1]

Example[]

Continuing with the example of the fractal string associated to the middle thirds Cantor set, we compute ${\displaystyle \zeta _{\mathbb {C} }(s)=\sum _{n=1}^{\infty }{\frac {2^{n-1}}{3^{ns}}}={\frac {\frac {1}{3^{s}}}{1-{\frac {2}{3^{s}}}}}={\frac {1}{3^{s}-2}}}$.^[1][2] We compute the abscissa of convergence to be the value of ${\displaystyle s}$ satisfying ${\displaystyle 3^{s}=2}$, so that ${\displaystyle s=\log _{3}2={\frac {\log 2}{\log 3}}}$ is the Minkowski dimension of the Cantor set.^{[citation needed]} For complex ${\displaystyle s}$, ${\displaystyle \zeta _{\mathbb {C} }(s)}$ has poles at the infinitely many solutions of ${\displaystyle 3^{s}=2}$, which, for this example, occur at ${\displaystyle s={\frac {\log 2+2\pi ik}{\log 3}}}$, for all integers ${\displaystyle k}$. This collection of points is called the set of complex dimensions of the middle thirds Cantor set.^[1][2]

Applications[]

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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.^{[citation needed]}

The existence of non-real complex dimensions with positive real part has been proposed^{[by whom?]} to be the signature feature of fractal objects.^[3] 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.^[3] This new definition of fractality solves some old problems in fractal geometry. For example, everyone^[who?] can agree that Cantor's devil's staircase is fractal, which it is with this new definition of fractality in terms of complex dimensions^[why?], but it is not in the sense of Mandelbrot.

Generalized fractal strings[]

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A generalized fractal string ${\displaystyle \eta }$ is defined to be a local positive or complex measure on ${\displaystyle (0,+\infty )}$ such that ${\displaystyle |\eta |(0,x_{0})=0}$ for some ${\displaystyle x_{0}>0}$, where the positive measure ${\displaystyle |\eta |}$ is the variation of ${\displaystyle \eta }$.^{[1][clarification needed]} For example, if ${\displaystyle {\mathcal {L}}=\{l_{j}\}_{j=1}^{\infty }}$ is an ordinary fractal string with multiplicities ${\displaystyle w_{j}}$, then the measure ${\displaystyle \eta _{\mathcal {L}}:=\sum _{j=1}^{\infty }w_{j}\delta _{l_{j}^{-1}}}$ associated to ${\displaystyle {\mathcal {L}}}$ (where ${\displaystyle \delta _{\{x\}}}$ refers to the Dirac delta measure concentrated at the point ${\displaystyle x}$) is an example of a generalized fractal string.^[1] Even more concretely, one may consider, for example, the generalized Cantor string ${\displaystyle \eta _{CS}:=\sum _{j=1}^{\infty }b^{j}\delta _{a^{j}}}$ for ${\displaystyle 1<b<a}$.^[1]

If ${\displaystyle \eta }$ is a generalized fractal string, then its dimension is defined as

$${\displaystyle D_{\eta }:=\inf(\sigma \in \mathbb {R} :\int _{0}^{\infty }x^{-\sigma }|\eta |(dx)<\infty ),}$$

$${\displaystyle N_{\eta }(x):=\int _{0}^{x}\eta (dx)=\eta (0,x)}$$

$${\displaystyle \zeta _{\eta }(s):=\int _{0}^{\infty }x^{-s}\eta (dx).}$$

References[]