15,507 research outputs found
Numerical investigations of discrete scale invariance in fractals and multifractal measures
Fractals and multifractals and their associated scaling laws provide a
quantification of the complexity of a variety of scale invariant complex
systems. Here, we focus on lattice multifractals which exhibit complex
exponents associated with observable log-periodicity. We perform detailed
numerical analyses of lattice multifractals and explain the origin of three
different scaling regions found in the moments. A novel numerical approach is
proposed to extract the log-frequencies. In the non-lattice case, there is no
visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set
of complex exponents spread irregularly within the complex plane. A non-lattice
multifractal can be approximated by a sequence of lattice multifractals so that
the sets of complex exponents of the lattice sequence converge to the set of
complex exponents of the non-lattice one. An algorithm for the construction of
the lattice sequence is proposed explicitly.Comment: 31 Elsart pages including 12 eps figure
On random fractals with infinite branching: definition, measurability, dimensions
We discuss the definition and measurability questions of random fractals and
find under certain conditions a formula for upper and lower Minkowski
dimensions. For the case of a random self-similar set we obtain the packing
dimension.Comment: 11 page
Spectral zeta functions of fractals and the complex dynamics of polynomials
We obtain formulas for the spectral zeta function of the Laplacian on
symmetric finitely ramified fractals, such as the Sierpinski gasket, and a
fractal Laplacian on the interval. These formulas contain a new type of zeta
function associated with a polynomial (rational functions also can appear in
this context). It is proved that this zeta function has a meromorphic
continuation to a half plain with poles contained in an arithmetic progression.
It is shown as an example that the Riemann zeta function is the zeta functions
of a quadratic polynomial, which is associated with the Laplacian on an
interval. The spectral zeta function of the Sierpinski gasket is a product of
the zeta function of a polynomial and a geometric part; the poles of the former
are canceled by the zeros of the latter. A similar product structure was
discovered by M.L. Lapidus for self-similar fractal strings
Dirac operators and spectral triples for some fractal sets built on curves
We construct spectral triples and, in particular, Dirac operators, for the
algebra of continuous functions on certain compact metric spaces. The triples
are countable sums of triples where each summand is based on a curve in the
space. Several fractals, like a finitely summable infinite tree and the
Sierpinski gasket, fit naturally within our framework. In these cases, we show
that our spectral triples do describe the geodesic distance and the Minkowski
dimension as well as, more generally, the complex fractal dimensions of the
space. Furthermore, in the case of the Sierpinski gasket, the associated
Dixmier-type trace coincides with the normalized Hausdorff measure of dimension
.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv.
Mat
Explicit Spectral Decimation for a Class of Self--Similar Fractals
The method of spectral decimation is applied to an infinite collection of
self--similar fractals. The sets considered belong to the class of nested
fractals, and are thus very symmetric. An explicit construction is given to
obtain formulas for the eigenvalues of the Laplace operator acting on these
fractals
An attractive critical point from weak antilocalization on fractals
We report a new attractive critical point occurring in the Anderson
localization scaling flow of symplectic models on fractals. The scaling theory
of Anderson localization predicts that in disordered symplectic two-dimensional
systems weak antilocalization effects lead to a metal-insulator transition.
This transition is characterized by a repulsive critical point above which the
system becomes metallic. Fractals possess a non-integer scaling of conductance
in the classical limit which can be continuously tuned by changing the fractal
structure. We demonstrate that in disordered symplectic Hamiltonians defined on
fractals with classical conductance scaling , for , the metallic phase is replaced
by a critical phase with a scale invariant conductance dependent on the fractal
dimensionality. Our results show that disordered fractals allow an explicit
construction and verification of the expansion
Laplace Operators on Fractals and Related Functional Equations
We give an overview over the application of functional equations, namely the
classical Poincar\'e and renewal equations, to the study of the spectrum of
Laplace operators on self-similar fractals. We compare the techniques used to
those used in the euclidean situation. Furthermore, we use the obtained
information on the spectral zeta function to define the Casimir energy of
fractals. We give numerical values for this energy for the Sierpi\'nski gasket
- âŠ