Michel L Lapidus

Michel L Lapidus
University of California, Riverside | UCR · Department of Mathematics

Ph.D., Doctorat es Sc., Habil. Distinguished Professor in Mathematics, Burton Jones Endowed Chair in Pure Mathematics

About

196
Publications
31,626
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3,514
Citations
Additional affiliations
July 1990 - present
University of California, Riverside
Position
  • Professor
Description
  • Professor of Mathematics; Cooperating Faculty member in Electrical Engineering, as well as in Computer Science and Engineering, and in the Department of Physics and Astronomy.
September 1986 - June 1990
University of Georgia
Position
  • Associate Professor of Mathematics
September 1984 - June 1985
Mathematical Sciences Research Institute
Mathematical Sciences Research Institute
Position
  • Member of the MSRI
Description
  • Member of the research program on Operator Algebras.
Education
March 1986 - June 1987
Sorbonne Université
Field of study
  • Mathematics
September 1984 - June 1986
Sorbonne Université
Field of study
  • Mathematics (and Mathematical Physics)
September 1979 - June 1980
University of California, Berkeley
Field of study
  • Postdoctoral (and George) Fellow in Mathematics

Publications

Publications (196)
Article
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 gasket is the limit of finite graphs consisting of various affine images of an eq...
Article
Full-text available
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...
Preprint
Full-text available
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 ad\`elic fractal strings in order to reveal the oscillatory nature of ad\`elic fr...
Preprint
Full-text available
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 Sierpi\'nski is the limit of finite graphs consisting of various affine images of an equilat...
Preprint
Full-text available
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...
Preprint
Full-text available
We study the essential singularities of geometric zeta functions $\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{\infty}<D_1\le D$, we construct a bounded fractal string $\mathcal L$ such that $D_{\rm par}(\zeta_{\mathcal L})=D_{\...
Article
Full-text available
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 \S\ref{Sec:2}, and then in the higher-dimensional case of relative fractal drums and, in par...
Chapter
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 in the case of curves and eventually, by Deligne in the case of varietie...
Chapter
In this chapter, we introduce the notion of relative fractal drums (or RFDs, in short). They represent a simple and natural extension of two fundamental objects of fractal analysis, simultaneously: that of bounded sets in ℝN (i.e., of fractals) and that of bounded fractal strings (introduced by the first author and Carl Pomerance in the early 1990s...
Chapter
In this chapter, we reconstruct information about the geometry of relative fractal drums (and, consequently, compact sets) in \(\mathbb{R}^{N}\) from their associated fractal zeta functions. Roughly speaking, given a relative fractal drum (A, Ω) in \(\mathbb{R}^{N}\) (with N ≥ 1 arbitrary), we derive an asymptotic formula for its relative tube func...
Chapter
In this last chapter, we first introduce a refinement of the classification of bounded sets in ℝN which had begun with the well-known distinction between Minkowski nondegenerate and Minkowski degenerate sets. Further distinction will be made by classifying fractals according to the properties of their tube functions and allowing, in particular, mor...
Chapter
Distance and tube zeta functions of fractals in Euclidean spaces can be considered as a bridge between the geometry of fractal sets and the theory of holomorphic functions. This is first seen from their fundamental property: the upper box dimension of any bounded fractal is equal to the abscissa of convergence of its distance and tube zeta function...
Chapter
In this chapter, we show that some fundamental geometric and number-theoretic properties of fractals can be studied by using their distance and tube zeta functions. This will motivate us, in particular, to introduce several new classes of fractals. Especially interesting among them are the transcendentally quasiperiodic sets, since they can be plac...
Article
Full-text available
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...
Book
This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal stri...
Article
Full-text available
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...
Article
Full-text available
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 subse...
Article
Full-text available
The 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 fr...
Article
We study meromorphic extensions of distance and tube zeta functions, as well as of zeta functions of fractal strings, which include perturbations of the Riemann zeta function. The distance zeta function $\zeta_A(s):=\int_{A_\delta} d(x,A)^{s-N}\mathrm{d}x$, where $\delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$, has b...
Article
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 $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\mathrm{d} x$ for all $s\in{\mathbb C}$ with $\operatorname{Re}\,s$ su...
Article
Full-text available
We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments...
Article
Full-text available
We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we examine the limiting behavior of particular sequences of compatible periodic orbits and, more interesting, in...
Article
Full-text available
This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c≥0, the spectral operator [Formula: see text] can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c. Rigorously, it turns...
Article
Full-text available
We construct Dirac operators and spectral triples for certain, not necessarily self-similar, fractal sets built on curves. Connes' distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami's measurable Riemannian geometry, which is a metric realization of the Sierpinski gas...
Article
Full-text available
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). O...
Article
Full-text available
The theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called “distance zeta functions,” whi...
Conference Paper
Full-text available
In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A clo...
Chapter
Full-text available
For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichl...
Article
Full-text available
We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \ci...
Article
Full-text available
If D is a rational polygon, then the associated rational billiard table is given by \Omega(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding fractal billiard table is denoted by \Omega(F). In this paper, we survey many of the results from [LapNie1-3]...
Article
Full-text available
In this survey article, we investigate the spectral properties of fractal differential operators on self-similar fractals. In particular, we discuss the decimation method, which introduces a renormalization map whose dynamics describes the spectrum of the operator. In the case of the bounded Sierpinski gasket, the renormalization map is a polynomia...
Book
From the Back Cover (and the Preface): This volume contains the proceedings from three conferences: the PISR 2011 International Conference on "Analysis, Fractal Geometry, Dynamical Systems and Economics", held November 8-12, 2011 in Messina, Siciliy, Italy, on the occasion of the 2011 Anassilaos International Research Prize in Mathematics, awarded...
Article
Full-text available
The multifractal structure underlying a self-similar measure stems directly from the weighted self-similar system (or weighted iterated function system) which is used to construct the measure. This follows much in the way that the dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity dimension, is determined by the scaling...
Article
Full-text available
We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the...
Article
Full-text available
The theory of p-adic fractal strings and their complex dimensions was developed by the first two authors in [17, 18, 19], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [28]. Using the fractal tube formula obtained by the authors for p-adic fractal strings in...
Article
Full-text available
A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. This problem is related to the question "Can one hear the shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer for fractal strings whose dimension is $c\in(...
Article
Full-text available
In a previous paper [arXiv:1006.3807, Adv. in Math. vol. 227, 2011, pp. 1349-1398], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypothese...
Book
From the Back Cover (and the Preface): This volume contains the proceedings from three conferences: the PISR 2011 International Conference on "Analysis, Fractal Geometry, Dynamical Systems and Economics", held November 8-12, 2011 in Messina, Siciliy, Italy, on the occasion of the 2011 Anassilaos International Research Prize in Mathematics, awarded...
Chapter
In this chapter we discuss new work motivated by the notion of complex dimension. Throughout, we also make numerous suggestions for the direction of future research related to, and naturally extending in various ways, the theory developed in this book. In several places, we also provide some additional background material that may be useful to the...
Chapter
In this chapter, we develop the notion of generalized fractal string, viewed as a measure on the half-line. This is more general than the notion of fractal string considered in Chapter 1 and in the earlier work on this subject (see the notes to Chapter 1).
Chapter
In this chapter, we apply our explicit formulas to obtain an asymptotic expansion for the prime orbit counting function of suspended flows. The resulting formula involves a sum of oscillatory terms associated with the dynamical complex dimensions of the flow.
Chapter
In this chapter we discuss some more philosophical aspects of our theory of complex dimensions. In Section 12.1, we propose a new definition of fractality, involving the notion of complex dimension.
Chapter
The study of the complex dimensions of nonlattice self-similar strings is most naturally carried out in the more general setting of Dirichlet polynomials.
Chapter
In this chapter we give various examples of explicit formulas for the counting function of the lengths and frequencies of (generalized) fractal strings and sprays.
Chapter
Throughout this book, we use an important class of ordinary fractal strings, the self-similar fractal strings, to illustrate our theory. These strings are constructed in the usual way via contraction mappings. In this and the next chapter, we give a detailed analysis of the structure of the complex dimensions of such fractal strings
Chapter
In this chapter, we obtain (in Section 8.1) a distributional formula for the volume of the tubular neighborhoods of the boundary of a fractal string, called a tube formula. In Section 8.1.1, under more restrictive assumptions, we also derive a tube formula that holds pointwise. In Section 8.3, we then deduce from these formulas a new criterion for...
Chapter
In this chapter, we recall some basic definitions pertaining to the notion of (ordinary) fractal string and introduce several new ones, the most important of which is the notion of complex dimension. We also give a brief overview of some of our results in this context by discussing the simple but illustrative example of the Cantor string. In the la...
Chapter
In this chapter, we analyze the oscillations in the geometry and the spectrum of the simplest type of generalized self-similar fractal strings. The complex dimensions of these generalized Cantor strings form a vertical arithmetic sequence D + inp (n Z).
Article
Full-text available
The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult,...
Article
Thèse (Doctorat Nouveau régime) -- 1986. Comprend une bibliographie. Microfiche.
Article
Full-text available
Abstract We investigate the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e. infinite) Sierpinski gasket and a self-similar Sturm–Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, Sabot discovered the relation between the spectrum of this oper...
Article
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For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topolog...
Article
Full-text available
The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present...
Article
We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).
Article
In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the frac...
Article
Full-text available
We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the tubular neighborhood of a p-a...
Article
Full-text available
We describe the periodic orbits of the prefractal Koch snowflake billiard (the nth inner rational polygonal approximation of the Koch snowflake billiard). In the case of the finite (prefractal) billiard table, we focus on the direction given by an initial angle of pi/3, and define 1) a compatible sequence of piecewise Fagnano orbits, 2) an eventual...
Article
Full-text available
ABSTRACT: In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard $KS$. This is a priori a very difficult problem because $\partial(KS)$, the snowflake curve boundary of $KS$, is nowhere differentiable, making it impossible to apply the usual law of reflection at any point of the boundary of the billiard...
Article
Full-text available
For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichl...
Article
Full-text available
We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This...
Article
Full-text available
We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notio...
Article
Full-text available
Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this tex...
Chapter
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The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.
Article
In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001) 591–620] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this...
Article
Full-text available
We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notio...