# Michel L LapidusUniversity of California, Riverside | UCR · Department of Mathematics

Michel L Lapidus

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

## About

196

Publications

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Introduction

Additional affiliations

September 1986 - June 1990

September 1984 - June 1985

**Mathematical Sciences Research Institute**

Position

- Member of the MSRI

Description

- Member of the research program on Operator Algebras.

Education

March 1986 - June 1987

September 1984 - June 1986

September 1979 - June 1980

## Publications

Publications (196)

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...

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...

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...

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...

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...

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_{\...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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]...

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...

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...

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...

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...

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...

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(...

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...

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...

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...

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).

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.

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.

The study of the complex dimensions of nonlattice self-similar strings is most naturally carried out in the more general setting of Dirichlet polynomials.

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.

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

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...

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...

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).

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,...

Thèse (Doctorat Nouveau régime) -- 1986. Comprend une bibliographie. Microfiche.

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...

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...

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...

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”).

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...