Fractal string information

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

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

An ordinary fractal string $\Omega$ 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 ${\mathcal {L}}=\{\ell _{1},\ell _{2},\ldots \}$ written in non-increasing order; we also refer to ${\mathcal {L}}$ as a fractal string. For example, ${\mathcal {L}}=\left\{{\frac {1}{3}},{\frac {1}{9}},{\frac {1}{9}},\ldots \right\}$ 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 $\Omega$ has a boundary $\partial \Omega$ 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 ${\mathcal {L}}$ of the set itself is "intrinsic," in the sense that the fractal string ${\mathcal {L}}$ itself (independent of a specific geometric realization of these lengths as corresponding to a choice of set $\Omega$ ) contains information about the fractal to which it corresponds.^[1]

For each fractal string ${\mathcal {L}}$ , we can associate to ${\mathcal {L}}$ a geometric zeta function $\zeta _{\mathcal {L}}$ : the Dirichlet series $\zeta _{\mathcal {L}}(s)=\sum _{j\in \mathbb {J} }\ell _{j}^{s}$ . Informally, the geometric zeta function carries geometric information about the underlying fractal, particularly in the location of its poles and the residues of the zeta function at these poles. These poles of (the analytic continuation of) the geometric zeta function $\zeta _{\mathcal {L}}(s)$ are then called complex dimensions of the fractal string ${\mathcal {L}}$ , and these complex dimensions appear in formulae which describe the geometry of the fractal.^[1]

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 $s={\frac {\log 2+2\pi ik}{\log 3}}$ , 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 $\eta$ is defined to be a local positive or complex measure on $(0,+\infty )$ such that $|\eta |(0,x_{0})=0$ for some $x_{0}>0$ , where the positive measure $|\eta |$ is the total variation measure associated to $\eta$ . 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.

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