Complex Fractal Dimensions

• Complex fractal dimensions are rigorously defined as the poles of associated zeta functions that refine classical fractal measures by encoding both scaling properties and oscillatory geometry.
• They underpin explicit tube formulas that link the volume of fractal neighborhoods with spectral and geometric properties, demonstrating applications in dynamics, network theory, and physics.
• Emerging research extends these notions to nonarchimedean and adelic contexts, forging deep connections with number theory, spectral analysis, and algorithmic complexity.

Complex fractal dimensions are a rigorous extension of classical fractal dimensions, defined as the poles (and, in maximal generality, more general singularities) of suitably constructed zeta functions associated with the geometry or dynamics of sets, measures, or networks. These complex dimensions serve as spectral fingerprints encoding not only the main scaling properties (through their real parts, often coinciding with classical box or Minkowski dimension), but also intricate geometric oscillations and self-similar structures (through their imaginary parts). Formulated through the language of Dirichlet and fractal zeta functions, this theory applies across Euclidean, nonarchimedean, and network-theoretic settings, and has far-reaching connections to spectral asymptotics, number theory (notably the Riemann Hypothesis), nonlinear dynamics, and statistical physics.

1. Foundational Zeta Functions and Definition of Complex Dimensions

The core methodology underlying complex fractal dimensions is the association of geometric (or spectral) zeta functions to a given object—be it a fractal string (a sequence of positive real numbers interpreted as lengths), a compact or relative subset of a metric space, a function graph, or a structure in nonarchimedean or network settings.

For a fractal string $\mathcal{L} = (\ell_j)_{j \geq 1}$, the geometric zeta function is

$\zeta_\mathcal{L}(s) = \sum_{j=1}^\infty \ell_j^s,$

convergent for $\mathrm{Re}(s)$ sufficiently large. For a compact set $A \subset \mathbb{R}^N$, the distance zeta function is

$\zeta_A(s) = \int_{A_\delta} d(x, A)^{s-N} dx$

for some fixed $\delta > 0$; the tube zeta function uses the parallel sets, integrating their volume: $$\widetilde\zeta_A(s) = \int_0^\delta t^{s-N-1} |A_t| dt.$$ In the $p$-adic (nonarchimedean) setting, the geometric zeta function of a fractal string is

$\zeta_{\mathcal{L}_p}(s) = \sum_j \mu_H(B_j)^s$

with $\mu_H$ the Haar measure and $B_j$ p-adic balls.

"Complex dimensions" are the poles (and possibly essential singularities in "maximally hyperfractal" cases) of a suitable meromorphic extension of these zeta functions beyond their abscissa of convergence. The abscissa itself always coincides with the Minkowski (or upper box) dimension in the classical setting, so that the set of complex dimensions, denoted e.g. $\mathcal{D}_A$, serves as a refinement of classical fractal dimension: $$\mathcal{D}_A = \{ \omega \in U \subset \mathbb{C} : \omega \text{ is a pole of } \zeta_A(s) \}.$$ The main pole at $s = D$ (the Minkowski/box dimension) is often simple, but additional nonreal complex dimensions encode log-periodic oscillations and deeper self-similar or quasiperiodic structure (Lapidus et al., 2015, Lapidus et al., 2015, Lapidus et al., 2016, Lapidus et al., 2014).

2. Tube Formulas, Oscillatory Terms, and Geometric Implications

A central analytic achievement is the derivation of explicit "fractal tube formulas" expressing the volume of a $t$-neighborhood (or tube) around a set (or relative fractal drum, RFD) as a sum over complex dimensions: $$|A_t| = \sum_{\omega \in \mathcal{D}_A} \operatorname{res}\left(\frac{t^{N-s}\zeta_A(s)}{N-s}, \omega\right) + R(t),$$ where $R(t)$ is an error term which vanishes under strong regularity (languidity) conditions (Lapidus et al., 2015, Lapidus et al., 2016, Lapidus et al., 2016, Lapidus, 2018).

For a complex dimension $\omega = \omega_0 + i \tau$, the associated tube term

$t^{N-\omega} = t^{N-\omega_0} \exp(-i \tau \log t)$

manifests as a power-law decay modulated by oscillations in $\log t$, with frequency set by $\operatorname{Im}(\omega)$. Thus, the spectrum of complex dimensions fully encodes both fractal scaling and oscillatory geometry. When the only pole on the "critical line" $\mathrm{Re}(s) = D$ is $s = D$ itself (and it is simple), one obtains precise Minkowski measurability (Dettmers et al., 2015, Lapidus et al., 2015). In the presence of further poles or higher order, nontrivial oscillatory behavior enters, precluding classical measurability and leading to log-periodic or multifractal structure (Lapidus et al., 2015, Lapidus et al., 2016).

3. Lattice/Nonlattice Dichotomy, Hyperfractality, and Maximal Complexity

A foundational dichotomy emerges in self-similar systems defined by contraction ratios $\{r_j\}$. In the "lattice" case (all ratios are integer powers of a common scaling $r$), complex dimensions repeat periodically along vertical lines in the complex plane, e.g.,

$$\mathcal{D} = \{ D + 2\pi i n / \log(1/r) : n \in \mathbb{Z} \}$$

and tube formulas exhibit log-periodic oscillations (Lapidus et al., 2011, Dettmers et al., 2015). In the "nonlattice" case, such periodicity is absent, typically yielding a single principal complex dimension on the critical line and asymptotically regular (measurable) behavior.

Extending these ideas, constructions such as transcendentally (possibly infinitely-) quasiperiodic sets, built via generalized Cantor sets with carefully chosen algebraic parameters, allow for the arrangement of complex dimensions with a dense irrational or even maximal (every point is a nonremovable singularity) distribution along the critical line. These "maximally hyperfractal" sets, whose zeta functions have a natural boundary along $\mathrm{Re}(s)=D$, exhibit the maximal degree of geometric oscillation and irreducible analytic complexity in their scaling behaviors (Lapidus et al., 2015, Lapidus et al., 2016, Lapidus et al., 2014).

4. Nonarchimedean, Adelic, and Spectral Connections

The notion of complex dimensions naturally generalizes to nonarchimedean and adelic settings. In $p$-adic fractal strings, lengths take the form $p^{-n}$ and the geometric zeta function becomes a rational function of $z = r^s$; all self-similar $p$-adic strings are "lattice," and their complex dimensions are thus periodically distributed along finitely many vertical lines (Lapidus et al., 2011, Lapidus et al., 2020). Explicit volume (tube) formulas in this setting express the $\epsilon$-neighborhood volumes as sums over residues at complex dimensions, paralleling the real case.

Adelic constructions—Cartesian products of $p$-adic and archimedean fractal strings—serve as a "global" unification, with the prospect of relating the distribution of complex dimensions to global aspects of the Riemann zeta function and the Riemann Hypothesis. For example, a self-similar $p$-adic string of rational Minkowski–Bouligand dimension $D = k/m$ has complex dimensions

$$\mathcal{D}_{L_p} = \left\{ D + i n \frac{2\pi}{m \log p} : n \in \mathbb{Z} \right\},$$

with the oscillatory period determined by $(m \log p)^{-1}$. Assembling these strings across all $p$ yields adelic fractal objects capturing both local and global oscillatory phenomena (Lapidus et al., 2020).

Spectral theory provides further deep connections. The spectral zeta function of a "fractal drum" (bounded domain with fractal boundary) or a Laplacian on a fractal structure admits meromorphic extension whose poles (spectral complex dimensions) govern Weyl–Berry asymptotics and spectral oscillations. This approach has produced an operator-theoretic reformulation of the Riemann Hypothesis: the spectral operator $\mathfrak{a}_c = \zeta(\partial_c)$ acting on a weighted Hilbert space is quasi-invertible for all $c \in (0,1)\setminus\{1/2\}$ if and only if $\zeta(s)$ has no zeros off the critical line—thus linking fractal geometry, spectrum, and analytic number theory (Herichi et al., 2012).

5. Applications in Dynamical Systems, Physics, and Networks

Complex fractal dimensions have broad applications beyond geometric measure theory. In deterministic and stochastic iterated graph systems, the Hausdorff and box-counting (Minkowski) dimensions can be explicitly computed and coincide; for deterministic models,

$$\dim_B(\Xi) = \dim_H(\Xi) = \frac{\log \rho(\mathbb{M})}{\log \rho_{\min}(\mathcal{D})}$$

where $\rho(\mathbb{M})$ and $\rho_{\min}(\mathcal{D})$ are the spectral radii of substitution matrices. In the random case, Lyapunov exponents provide an analogous characterization. These results offer a rigorous bridge between fractal geometry and graph/network theory, with implications for modeling self-similar complex networks, modularity, and robustness under random growth (Li, 2022, Makulski et al., 6 Feb 2025).

Topological data analysis (TDA), particularly persistent homology–based and magnitude-based approaches, provides tools for estimating and interpreting fractal dimensions of networks. PH dimension and magnitude dimension estimators exploit higher-order topological features, clique complexes, and filtered simplicial complexes to robustly capture self-similarity and higher-order interactions in networks, offering conceptual and computational advantages over classical box-counting (Andreeva et al., 18 Jun 2025).

In nonlinear PDEs and physics, explicit fractal solutions exhibit noninteger box-counting or Hausdorff dimension, measured via voxel-based or box-counting techniques and confirmed through multi-scale self-affine visualization—directly linking complex dimension theory with spatial complexity of analytic or physical models (e.g., in turbulent fluid or plasma systems) (Dutta et al., 3 Aug 2025).

The heat content of domains with self-similar fractal boundaries is asymptotically expanded in powers determined by the complex dimensions of the underlying geometry. Through the method of Mellin transforms applied to scaling functional equations, these exponents (complex dimensions) govern both the leading and oscillatory terms in, for instance, heat diffusion on fractal domains (Hoffer et al., 13 Aug 2025).

6. Computational, Algorithmic, and Information-theoretic Aspects

Complex dimensions connect with algorithmic information theory via pointwise and setwise Kolmogorov complexity. The algorithmic dimension of a point—measured via the limiting density of algorithmic information—can be used to derive intersection and product formulae for Hausdorff and packing dimensions in arbitrary sets, not only in Borel or analytic regularity settings. This algorithmic perspective facilitates proofs and generalizations for classical dimension inequalities using pointwise complexity and oracles, emphasizing the local nature of fractal structure (Lutz, 2016).

Methods for determining the fractal dimension of function graphs—especially for complex-valued or vector-valued fractal functions—exploit the invariance of Hausdorff and box dimensions under Lipschitz (or bi-Lipschitz) mappings from the graph to lower-dimensional projections. The self-affine nature and richer structure of graphs of complex-valued functions, especially those constructed via iterated function systems with nontrivial "germ," "base," and complex scaling functions, provide sharper bounds and show that classical results for real-valued functions do not generalize naively (Verma et al., 2022).

7. Open Problems and Future Directions

Research continues in several prominent directions:

• The extension of fractal complex dimension theory to higher dimensions, to non-Euclidean and nonarchimedean settings (including Berkovich spaces), and to more general dynamical systems.
• The investigation of maximal hyperfractals and transcendentally quasiperiodic complexes, both in geometric and spectral settings (Lapidus et al., 2015, Lapidus et al., 2014).
• The characterization of multifractal and maximally oscillatory phenomena—in both geometric and spectral contexts—via explicit tube or heat content formulas.
• Application of zeta function techniques to new classes of networks and temporal evolution, including dynamical graph models, percolation on fractal structures, and dynamic processes such as random walks.
• The pursuit of a unified, potentially adelic theory of complex dimensions tightly linked to the analytic properties of global zeta and $L$-functions, and new spectral reformulations of deep conjectures in analytic number theory (Lapidus et al., 2020, Herichi et al., 2012).
• TDA-based and magnitude-based approaches for fractal dimension estimation in networks, together with improvements in computational efficiency and interpretability (Andreeva et al., 18 Jun 2025).

The development and deployment of complex fractal dimensions continue to provide detailed and robust bridges across geometric measure theory, spectral analysis, dynamics, number theory, and real-world applications ranging from nonlinear wave propagation to the architecture of complex networks.