Fractal geometry, complex dimensions and zeta functions : geometry and spectra of fractal strings

583 Pages·2013·3.18 MB·English

by  Michel L Lapidus

Springer Monographs in Mathematics

Forfurther volumes:
http://www.springer.com/series/3733

Michel L. Lapidus • Machiel van Frankenhuijsen
Fractal Geometry,
Complex Dimensions
and Zeta Functions
Geometry and Spectra of Fractal Strings
Second Edition
With 73 illustrations
Michel L. Lapidus Machiel van Frankenhuijsen
Department of Mathematics Department of Mathematics
University of California, Riverside Utah Valley University
Riverside, CA, USA Orem, UT, USA

The front cover shows a tubular neighborhood of the Devil’s staircase (Figure 12.2,
page 338) and the quasiperiodic pattern of the complex dimensions of a nonlattice self-
similar string (Figure 3.7, page 88).

ISSN 1439-7382
ISBN 978-1-4614-2175-7 ISBN 978-1-4614-2176-4 (eBook)
DOI 10.1007/978-1-4614-2176-4
Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012948283

Mathematics Subject Classification (2010): Primary: 11M26, 11M41, 14G10, 28A12, 28A75,
28A80, 35P20, 37C30, 37C45, 58J32; Secondary: 11G20, 11J70, 11M06, 11N05, 26E30, 30D35,
37B10, 58B34, 81Q20

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To my Riemann zeros:
Odile, Julie and Michaël, my muse and offspring
Michel L. Lapidus
To Jena, David and Samuel
Machiel van Frankenhuijsen

Overview
In this book, we develop a theory of complex dimensions of fractal strings
(i.e.,one-dimensionaldrumswithfractalboundary).Thesecomplexdimen-
sions are defined as the poles of the corresponding (geometric or spectral)
zeta function. They describe the oscillations in the geometry or the fre-
quency spectrum of a fractal string by means of an explicit formula. Such
oscillations are not observed in smooth geometries.
Along-termobjectiveofthisworkistomergeaspectsoffractal,spectral,
and arithmetic geometries. From this perspective, the theory presented in
thisbookenablesustoputthetheoryofDirichletseries(andofotherzeta
functions) in the geometric setting of fractal strings. It also allows us to
view certain fractal geometries as arithmetic objects by applying number-
theoreticmethodstothestudyofthegeometryandthespectrumoffractal
strings.
In Chapter 1, we first give an introduction to fractal strings and their
spectrum, and we precisely define the notion of complex dimension. We
then make in Chapter 2 an extensive study of the complex dimensions of
self-similar fractal strings. This study provides a large class of examples
to which our theory can be applied fruitfully. In particular, we show in
Chapter 3 that self-similar strings always have infinitely many complex
dimensions with positive real part, and that their complex dimensions are
quasiperiodicallydistributed.Thisisestablishedbyprovingthatthelattice
strings—thecomplexdimensionsofwhichareshowntobeperiodicallydis-
tributed along finitely many vertical lines—are dense (in a suitable sense)
inthesetofallself-similarstrings.WepresentthetheoryofChapter3—in
which we analyze in detail the quasiperiodic pattern of complex dimen-
vii
viii Overview
sions, via Diophantine approximation—by using the more general notion
of Dirichlet polynomial.
In Chapter 4, we extend the notion of fractal string to include (pos-
sibly virtual) geometries that are needed later on in our work. Then, in
Chapter 5, we establish pointwise and distributional explicit formulas (ex-
plicit in the sense of Riemann’s original formula [Rie1], but more general),
which should be considered as the basic tools of our theory. In Chapter 6,
we apply our explicit formulas to construct the spectral operator, which
expresses the spectrum in terms of the geometry of a fractal string. This
operator has an Euler product that is convergent, in a suitable sense to be
explained in Section 6.3.2, in the critical strip 0 < Res < 1 of the Rie-
mann zeta function. We also illustrate our formulas by studying a number
of geometric and direct spectral problems associated with fractal strings.
In Chapter 7, we use the theory of Chapters 3 and 5 to study a class of
suspended flows and define the associated dynamical complex dimensions.
Inparticular,weestablishanexplicitformulafortheperiodicorbitcounting
functionofsuchflowsanddeducefromitaprimeorbittheoremwithsharp
errortermforself-similarflows,therebyextendinginthiscontextthework
of [ParrPol1, 2]. We also obtain an Euler product for the zeta function of
a self-similar fractal string (or flow).
In Chapter 8, we derive pointwise and distributional explicit formulas
for the volume of the tubular neighborhoods of the boundary of a frac-
tal string. We deduce a new criterion for the Minkowski measurability of
a fractal string, in terms of its complex dimensions, extending the earlier
criterion obtained by the first author and C. Pomerance (see [LapPo2]).
This fractal tube formula suggests analogies with aspects of Riemannian
and convex geometry, thereby giving substance to a geometric interpreta-
tion of the complex dimensions. We also provide a detailed discussion of
the pointwise fractal tube formulas in the important special case of self-
similarstrings,andstudytheirconsequencesbothforlatticeandnonlattice
strings. In particular, we deduce that a self-similar fractal string is Min-
kowski measurable if and only if it is nonlattice (i.e., if the logarithms of
itsscalingratiosarenotpairwiserationallydependent).Wethenexplicitly
calculate the Minkowski content of a nonlattice string and the (suitably
defined) average Minkowski content of a lattice string.
In the later chapters of this book, Chapters 9–11, we analyze the con-
nections between oscillations in the geometry and the spectrum of fractal
strings. Thusweplacethespectral reformulation oftheRiemannhypothe-
sis, obtained by the first author and H. Maier [LapMa2], in a broader and
more conceptual framework, which applies to a large class of zeta func-
tions, including all those for which one expects the generalized Riemann
hypothesis to hold. We also reprove—and extend to a large subclass of the
aforementioned class—Putnam’stheorem accordingtowhich theRiemann
zetafunctiondoesnothaveaninfinitesequenceofcriticalzerosinarithme-
tic progression. Thiswork issupplementedin Section 11.1.1 with anupper
bound for the possible length of an arithmetic progression of zeros, and in
Section 11.4.1, where we present Mark Watkins’ work on the finiteness of
Overview ix
shifted arithmetic progressions of zeros of L-series. Based on these results,
wemakeanumberofconjecturesinSection11.5aboutthelocationofzeros
and poles of Dirichlet series.
The book culminates in Chapter 12 with a proposal for a new definition
of fractality as the presence of nonreal complex dimensions with positive
realpart.Accordingly,forexample,smoothRiemanniangeometriesarenot
fractal, but self-similar geometries are fractal, as expected. Furthermore,
the Devil’s staircase (the graph of the Cantor curve) is fractal in our sense
whereas it is not fractal according to Mandelbrot’s definition of fractality.
Moreover, arithmetic geometries should be fractal in this extended sense
(with the zeros and the poles of the associated arithmetic zeta functions
or L-functions playing the role of the complex dimensions). We elaborate
this definition and its consequences in several directions, including self-
similarity, number theory and arithmetic geometry, fractal cohomology,
and the spectrum of fractal drums.
InChapter13(andinSection12.2.1),wepresentnewworkon(ormotiv-
atedby)thetheoryofcomplexdimensions,aswellasmakeseveralsugges-
tions for future research in this area. In Section 12.2.1, we summarize the
resultsof[LapPe1]onthecomplexdimensionsandthevolumeofthetubu-
lar neighborhoods of the von Koch snowflake curve, which provides a first
example of a higher-dimensional theory of complex dimensions of fractals.
Furthermore, in Section 13.1, we survey the higher-dimensional theory of
complexdimensionsdevelopedbyE.Pearse,thefirstauthorandS.Winter
in [Pe2,LapPe2–3, LapPeWi1], via fractal tube formulas and tubular zeta
functions for fractal sprays and self-similar tilings. In Section 13.2, based
on the work of H. Lu and the authors in [LapLu1–3, LapLu-vF1–2], we
discuss the nonarchimedean analogue of aspects of the theory developed
in this book, via zeta functions for p-adic fractal strings and appropri-
ate nonarchimedean tube formulas. In Section 13.3, based on the work of
J. Rock, the first author and J. Lévy-Véhel in [LapRo1–3, LapLévRo,Ell-
LapMR],wediscussthebeginningofatheoryofmultifractalzetafunctions
which provides more refined topological and geometric information, both
in the context of fractal strings and of multifractal measures. Moreover, in
Section 13.4, we survey the work of B. Hambly and the first author [Ham-
Lap] on random fractal strings, the associated random zeta functions and
random fractal tube formulas. Finally, in Section 13.5, we briefly discuss
aspects of the theory of quantized fractal strings (or ‘fractal membranes’)
and their associated partition functions, as introduced by the first author
in [Lap9–10] and further developed with R. Nest in [LapNes1–3].