2,485 research outputs found
Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings
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 the rigorous functional
analytic framework given by the authors in [HerLa1] and within which to study
the spectral operator. Furthermore, we also give a necessary and sufficient
condition for the invertibility of the spectral operator (in the critical
strip) and therefore obtain a new spectral and operator-theoretic reformulation
of the Riemann hypothesis. More specifically, we show that the spectral
operator is invertible (or equivalently, that zero does not belong to its
spectrum) if and only if the Riemann zeta function zeta(s) does not have any
zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the
mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for
all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is
true. We also show the existence of four types of (mathematical) phase
transitions occurring for the spectral operator at the critical fractal
dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness,
its invertibility as well as its quasi-invertibility
Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator
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(0,1)-\{1/2} if and only if the
Riemann hypothesis is true. Later on, the spectral operator was introduced
heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of
complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of
a fractal string onto its spectrum. We focus here on presenting the rigorous
results obtained by the authors in [HerLa1] about the invertibility of the
spectral operator. We show that given any , the spectral operator
, now precisely defined as an unbounded normal
operator acting in a Hilbert space , is `quasi-invertible'
(i.e., its truncations are invertible) if and only if the Riemann zeta function
does not have any zeroes on the line . It follows
that the associated inverse spectral problem has a positive answer for all
possible dimensions , other than the mid-fractal case when ,
if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M.
van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc.,
Providence, RI, 2013. arXiv admin note: substantial text overlap with
arXiv:1203.482
Recent advances in open billiards with some open problems
Much recent interest has focused on "open" dynamical systems, in which a
classical map or flow is considered only until the trajectory reaches a "hole",
at which the dynamics is no longer considered. Here we consider questions
pertaining to the survival probability as a function of time, given an initial
measure on phase space. We focus on the case of billiard dynamics, namely that
of a point particle moving with constant velocity except for mirror-like
reflections at the boundary, and give a number of recent results, physical
applications and open problems.Comment: 16 pages, 1 figure in six parts. To appear in Frontiers in the study
of chaotic dynamical systems with open problems (Ed. Z. Elhadj and J. C.
Sprott, World Scientific
Sums of two dimensional spectral triples
We study countable sums of two dimensional modules for the continuous complex
functions on a compact metric space and show that it is possible to construct a
spectral triple which gives the original metric back. This spectral triple will
be finitely summable for any positive parameter. We also construct a sum of two
dimensional modules which reflects some aspects of the topological dimensions
of the compact metric space, but this will only give the metric back
approximately. We make an explicit computation of the last module for the unit
interval. The metric is recovered exactly, the Dixmier trace induces a multiple
of the Lebesgue integral and the number N(K) of eigenvalues bounded by K
behaves, such that N(K)/K is bounded, but without limit for K growing.Comment: 27 page
Fractal Strings and Multifractal Zeta Functions
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 topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
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