21 research outputs found
Fractality and Lapidus zeta functions at infinity
We study fractality of unbounded sets of finite Lebesgue measure at infinity
by introducing the notions of Minkowski dimension and content at infinity. We
also introduce the Lapidus zeta function at infinity, study its properties and
demonstrate its use in analysis of fractal properties of unbounded sets at
infinity.Comment: 19 pages, 1 figur
Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
In this paper we introduce an interesting family of relative fractal drums
(RFDs in short) at infinity and study their complex dimensions which are
defined as the poles of their associated Lapidus (distance) fractal zeta
functions introduced in a previous work by the author.
We define the tube zeta function at infinity and obtain a functional equation
connecting it to the distance zeta function at infinity much as in the
classical setting. Furthermore, under suitable assumptions, we provide general
results about existence of meromorphic extensions of fractal zeta functions at
infinity in the Minkowski measurable and nonmeasurable case. We also provide a
sufficiency condition for Minkowski measurability as well as an upper bound for
the upper Minkowski content, both in terms of the complex dimensions of the
associated RFD.
We show that complex dimensions of quasiperiodic sets at infinity posses a
quasiperiodic structure which can be either algebraic or transcedental.
Furthermore, we provide an example of a maximally hyperfractal set at infinity
with prescribed Minkowski dimension, i.e., a set such that the abscissa of
convergence of the corresponding fractal zeta function is in fact its natural
boundary.Comment: 31 pages, 2 figure
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 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 , we mean an explicit expression for the
volume of the -neighborhood of intersected by as a sum of
residues of a suitable meromorphic function (here, a fractal zeta function)
over the complex dimensions of the RFD . 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 Sierpi\'nski 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 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'.Comment: 90 pages (because of different style file), 5 figures, corrected
typos, updated reference
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
, where is
fixed and denotes the Euclidean distance from to extends the
definition of the zeta function associated with bounded fractal strings to
arbitrary bounded subsets of . The abscissa of Lebesgue
convergence coincides with , the upper box
dimension of . The complex dimensions of are the poles of the
meromorphic continuation of the fractal zeta function of to a suitable
connected neighborhood of the "critical line" . We establish
several meromorphic extension results, assuming some suitable information about
the second term of the asymptotic expansion of the tube function as
, where is the Euclidean -neighborhood of . We pay
particular attention to a class of Minkowski measurable sets, such that
as , with , and to a
class of Minkowski nonmeasurable sets, such that as , where is a nonconstant periodic
function and . In both cases, we show that can be
meromorphically extended (at least) to the open right half-plane
. Furthermore, up to a multiplicative constant, the
residue of evaluated at is shown to be equal to
(the Minkowski content of ) and to the mean value of (the average
Minkowski content of ), 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
.Comment: 30 pages, 2 figures, improved parts of the paper and shortened the
paper by reducing background material, to appear in Journal of mathematical
analysis and applications in 201
Essential singularities of fractal zeta functions
We study the essential singularities of geometric zeta functions
, associated with bounded fractal strings . For
any three prescribed real numbers , and in , such
that , we construct a bounded fractal string
such that , and . Here,
is the abscissa of absolute convergence of
, is the abscissa of
meromorphic continuation of , while is the infimum of all positive real numbers
such that is holomorphic in the open right half-plane
, except for possible isolated singularities in this
half-plane. Defining as the disjoint union of a sequence of
suitable generalized Cantor strings, we show that the set of accumulation
points of the set of essential singularities of , contained in the open right half-plane ,
coincides with the vertical line . We extend this
construction to the case of distance zeta functions of compact sets
in , for any positive integer .Comment: Theorem 3.2 (b) was wrong in the previous version, so we have decided
to omit it and pursue this issue at some future time. Part (b) of Theorem
3.2. was not used anywhere else in the paper. Theorem 3.2. is now called
Proposition 3.2. on page 12. Corrected minor typos and added new references
To appear in: Pure and Applied Functional Analysis; issue 5 of volume 5
(2020