arXiv subjects

Michel L. Lapidus

Publications and source records attributed to Michel L. Lapidus.

At least 19 recordsLinked to original sources

arXiv2026

Distributional Fractional Taylor Series and Interpretation of the Fractal and Number-Theoretic Explicit Formulas

Motivated by the work of M. L. Lapidus and M. van Frankenhuisjen on fractal explicit formulas for generalized fractal strings, we develop in this paper a distributional fractal Taylor's formula with error term and an exact distributional fractal Taylor series representation for generalized fractal strings. The explicit formulas that the aforementioned authors developed provide a way of expressing a generalized fractal string $η$ in terms of the underlying fractal complex dimensions, under suitable growth assumptions on its geometric zeta function, $ζ_η$, known as the languidity assumptions. This involves a sum indexed by the complex dimensions of the generalized fractal string, summing over all residues of $ζ_η$, multiplied by the Mellin transform of a tempered or Schwartz test function $ϕ$, denoted by $\widetildeϕ$. Under languidity assumptions, there is an error term present. Under stronger assumptions, that is, in the case of strong languidity, no error term is present and the resulting fractal explicit formula is said to be exact. In this paper, we express the residue term in the fractional distributional explicit formulas as the fractional $(-ω)^{\mathrm{th}}$-ordered distributional derivative of the Dirac $δ$ distribution, denoted by $Y_ω$, applied to modified test functions of the form $ϕ(x) \ln^{k-1}(x)$, where $k$ runs from $1$ to the multiplicity of the pole $ω$ of $ζ_η$. The coefficients in this sum are written in terms of the coefficients in the principal part of the Laurent expansion at $ω$ of the geometric zeta function $ζ_η$, multiplied by the gamma function evaluated at $ω$. The results obtained in this paper contribute to the broader program of characterizing fractals in terms of fractal Taylor series expansions involving their underlying fractal complex dimensions.

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arXiv2026

Emergent Loewner Dynamics in Slime Mold Growth

Growth fronts of slime molds are characterized through a direct geometric analysis based on Loewner evolutions, using experimentally acquired time-resolved images. The associated Loewner driving functions reconstructed from expanding pseudopod boundaries display statistical properties consistent with Gaussian-like behavior. A geometric estimate of the diffusivity parameter~$κ$ is inferred from fractal scaling, while Brownian diagnostics are assessed on the reconstructed driving signal. These findings show that the boundaries of a growing living organism display statistical and geometric properties consistent with emergent Loewner dynamics over experimentally accessible scales. This study establishes a quantitative framework for analyzing biological growth interfaces and suggests new connections between morphogenesis, stochastic geometry, and network reorganization under varying environmental conditions. We provide, to our knowledge, the first explicit reconstruction of a Loewner driving function from a living growth interface, revealing an emergent Brownian-like conformal growth regime at expanding fronts.

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arXiv2025

Fractal Zeta Functions and Complex Dimensions of Ahlfors Metric Measure Spaces

While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of these spaces. In this paper we will focus on the geometry of Ahlfors regular spaces, Metric Measure spaces with an additional regularity condition. Historically, fractals have been studied using different ideas of dimension which have all proven to be unsatisfactory to some degree. The theory of complex dimensions, developed by M.L. Lapidus and a number of collaborators, was developed in part to better understand fractality in the Euclidean case and seeks to overcome these problems. Of particular interest is the recent theory of complex dimensions in higher-dimensional Euclidean spaces, as studied by M.L. Lapidus, G. Radunović, and D. Zubrinić, who introduced and studied the properties of the distance zeta function $ζ_A$. We will show that this theory of complex dimensions naturally generalizes to the case of Ahlfors regular spaces, as the distance zeta function can be modified to these spaces and all of its main properties carry over. We also provide a selection of examples in Ahlfors spaces, as well as hints that the theory can be expanded to a more general setting.

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arXiv2025

On Complex Dimensions and Heat Content of Self-Similar Fractals

Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allow for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and in particular extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting.

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arXiv2024

Diffraction measures and patterns of the complex dimensions of self-similar fractal strings. I. The lattice case

We give a generalization of Lagarias' formula for diffraction by ideal crystals, and we apply it to the lattice case, in preparation for addressing the problem of quasicrystals and complex dimensions posed by Lapidus and van Frankenhuijsen concerning the quasiperiodic properties of the set of complex dimensions of any nonlattice self-similar fractal string. More specifically, in this paper, we consider the case of the complex dimensions of a lattice (rather than of a nonlattice) self-similar string and show that the corresponding diffraction measure exists, is unique, and is given by a suitable $\textit{continuous}$ analogue of a discrete Dirac comb. We also obtain more general results concerning the autocorrelation measures and diffraction measures of generalized idealized fractals associated to possibly degenerate lattices and the corresponding extension of the Poisson Summation Formula.

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arXiv2021

Metric Approximations of Spectral Triples on the Sierpiński Gasket and other fractal curves

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpiński is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.

math.OA↗

arXiv2020

$p$-adic fractal strings of arbitrary rational dimensions and Cantor strings

The local theory of complex dimensions for real and $p$-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for adèlic fractal strings in order to reveal the oscillatory nature of adèlic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar $p$-adic fractal strings of any rational dimension in the closed unit interval $[0,1]$. Moreover, as a first step towards a global theory of complex dimensions for adèlic fractal strings, we construct an adèlic Cantor string in the set of finite adèles $\mathbb{A}_0$ as an infinite Cartesian product of every $p$-adic Cantor string, as well as an adèlic Cantor-Smith string in the ring of adèles $\mathbb{A}$ as a Cartesian product of the general Cantor string and the adèlic Cantor string.

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arXiv2020

Quasiperiodic patterns of the complex dimensions of nonlattice self-similar strings, via the LLL algorithm

The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász.

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arXiv2020

Essential singularities of fractal zeta functions

We study the essential singularities of geometric zeta functions $ζ_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{\infty} α\}$, except for possible isolated singularities in this half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{\infty}$ of essential singularities of $ζ_{\mathcal L}$, contained in the open right half-plane $\{{\rm Re}\, s>D_{\infty}\}$, coincides with the vertical line $\{{\rm Re}\, s=D_{\infty}\}$. We extend this construction to the case of distance zeta functions $ζ_A$ of compact sets $A$ in $\mathbb{R}^N$, for any positive integer $N$.

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arXiv2018

Minkowski dimension and explicit tube formulas for $p$-adic fractal strings

The local theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a $p$-adic fractal string $\mathcal{L}_p$, expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a $p$-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and $p$-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

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arXiv2018

An Overview of Complex Fractal Dimensions: From Fractal Strings to Fractal Drums, and Back

Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), in \S2, and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of $\mbr^N$, for any integer $N \geq 1$, in \S3. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs, for which we refer to the author's previous (joint) books mentioned in the paper. Finally, in an epilogue (\S4), entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case) [HerLap1], along with [Lap8], and with T. Cobler (in the complex case) [CobLap1], respectively, as well as in the latter part of a book in preparation by the author, [Lap10].

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arXiv2018

Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measurability criterion established here is a generalization of the corresponding one obtained for fractal strings by the first author and M.\ van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion established here is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function. We also reflect on the notion of gauge-Minkowski measurability of RFDs and establish several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.

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arXiv2018

Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality

We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,Ω)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,Ω)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $Ω$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,Ω)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpiński gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $\mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devil's staircase) RFD, which is shown to be `subcritically fractal'.

math-ph↗

arXiv2017

Towards a fractal cohomology: Spectra of Polya--Hilbert operators, regularized determinants and Riemann zeros

Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil for curves and later, by Deligne for varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to those of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis is still on. In this paper, we study the properties of the derivative operator $D = \frac{d}{dz}$ on a particular weighted Bergman space of entire functions. The operator $D$ can be naturally viewed as the `infinitesimal shift of the complex plane'. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated to any suitable meromorphic function. We then show that the meromorphic function can be recovered by using a regularized determinant involving the above operator. This is illustrated in some important special cases: rational functions, zeta functions of curves over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. Our construction is motivated in part by [23] on the infinitesimal shift of the real line, as well as by earlier work of Deninger [10] on cohomology in number theory and a conjectural `fractal cohomology theory' envisioned in [25] and [28].

math.NT↗

arXiv2017

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.

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arXiv2017

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $ζ_A(s):=\int_{A_δ} d(x,A)^{s-N}\mathrm{d}x$, where $δ>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $\mathbb{R}^N$. The abscissa of Lebesgue convergence $D(ζ_A)$ coincides with $D:=\overline\dim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the "critical line" $\{\Re(s)=D\}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $t\to0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(\mathcal M+O(t^γ))$ as $t\to0^+$, with $γ>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^γ))$ as $t\to0^+$, where $G$ is a nonconstant periodic function and $γ>0$. In both cases, we show that $ζ_A$ can be meromorphically extended (at least) to the open right half-plane $\{\Re(s)>D-γ\}$. Furthermore, up to a multiplicative constant, the residue of $ζ_A$ evaluated at $s=D$ is shown to be equal to $\mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line $\{\Re(s)=D\}$.

math-ph↗

arXiv2016

Distance and tube zeta functions of fractals and arbitrary compact sets

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by $ζ_A(s)=\int_{A_δ}d(x,A)^{s-N}\,\mathrm{d} x$ for all $s\in\mathbb{C}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_δ$ is the $δ$-neighborhood of $A$, where $δ$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $ζ_A$ is equal to $\overline\dim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $ζ_A$ located on the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $ζ_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tildeζ_A(s)=\int_0^δ t^{s-N-1}|A_t|\,\mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tildeζ_A$ computed at $D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.

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arXiv2016

Zeta Functions and Complex Dimensions of Relative Fractal Drums: Theory, Examples and Applications

In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions', associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions' of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums' (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $\infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {\em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $m\ge1$, and even an infinite sequence of essential singularities along the critical line.

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