1,210 research outputs found
Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure
For directed graph iterated function systems (IFSs) defined on R, we prove
that a class of 2-vertex directed graph IFSs have attractors that cannot be the
attractors of standard (1-vertex directed graph) IFSs, with or without
separation conditions. We also calculate their exact Hausdorff measure. Thus we
are able to identify a new class of attractors for which the exact Hausdorff
measure is known
Differentiability of fractal curves
While self-similar sets have no tangents at any single point, self-affine
curves can be smooth. We consider plane self-affine curves without double
points and with two pieces. There is an open subset of parameter space for
which the curve is differentiable at all points except for a countable set. For
a parameter set of codimension one, the curve is continuously differentiable.
However, there are no twice differentiable self-affine curves in the plane,
except for parabolic arcs
Overlapping iterated function systems on a segment
Overlapping iterated function systems generate families of injective mappings from the attractor onto shift-invariant subsets of the code space. In this paper we consider an example of such a family for the uniformly linear systems of iterated functions on the unit segment. © Allerton Press, Inc., 2012
Recognition of Two-dimensional Shapes Based on Dependence Vectors
The aim of this paper is to present a new method of two-dimensional shape recognition. The method is based on dependence vectors which are fractal features extracted from the partitioned iterated function system. The dependence vectors show the dependency between range blocks used in the fractal compression. The effectiveness of our method is shown on four test databases. The first database was created by the authors and the other ones are: MPEG7 CE-Shape-1PartB, Kimia-99, Kimia-216. Obtained results have shown that the proposed method is better than the other fractal recognition methods of two-dimensional shapes
Inter-Intra Molecular Dynamics as an Iterated Function System
The dynamics of units (molecules) with slowly relaxing internal states is
studied as an iterated function system (IFS) for the situation common in e.g.
biological systems where these units are subjected to frequent collisional
interactions. It is found that an increase in the collision frequency leads to
successive discrete states that can be analyzed as partial steps to form a
Cantor set. By considering the interactions among the units, a self-consistent
IFS is derived, which leads to the formation and stabilization of multiple such
discrete states. The relevance of the results to dynamical multiple states in
biomolecules in crowded conditions is discussed.Comment: 7 pages, 7 figures. submitted to Europhysics Letter
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions
We consider the "Mandelbrot set" for pairs of complex linear maps,
introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and
others. It is defined as the set of parameters in the unit disk such
that the attractor of the IFS is
connected. We show that a non-trivial portion of near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of are in the closure of the set of interior points of ). Next we
turn to the attractors themselves and to natural measures
supported on them. These measures are the complex analogs of
much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os
and Garsia, we demonstrate how certain classes of complex algebraic integers
give rise to singular and absolutely continuous measures . Next we
investigate the Hausdorff dimension and measure of , for
in the set , for Lebesgue-a.e. . We also obtain partial results on
the absolute continuity of for a.e. of modulus greater
than .Comment: 22 pages, 5 figure
Equilibrium states and invariant measures for random dynamical systems
Random dynamical systems with countably many maps which admit countable
Markov partitions on complete metric spaces such that the resulting Markov
systems are uniformly continuous and contractive are considered. A
non-degeneracy and a consistency conditions for such systems, which admit some
proper Markov partitions of connected spaces, are introduced, and further
sufficient conditions for them are provided. It is shown that every uniformly
continuous Markov system associated with a continuous random dynamical system
is consistent if it has a dominating Markov chain. A necessary and sufficient
condition for the existence of an invariant Borel probability measure for such
a non-degenerate system with a dominating Markov chain and a finite (16) is
given. The condition is also sufficient if the non-degeneracy is weakened with
the consistency condition. A further sufficient condition for the existence of
an invariant measure for such a consistent system which involves only the
properties of the dominating Markov chain is provided. In particular, it
implies that every such a consistent system with a finite Markov partition and
a finite (16) has an invariant Borel probability measure. A bijective map
between these measures and equilibrium states associated with such a system is
established in the non-degenerate case. Some properties of the map and the
measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on
page 4 (the complete removal of the paragraph became the condition for the
publication in the DCDS-A after the reviewer ran out of the citation
suggestions collected in the paragraph
Fractal Dimensions in Perceptual Color Space: A Comparison Study Using Jackson Pollock's Art
The fractal dimensions of color-specific paint patterns in various Jackson
Pollock paintings are calculated using a filtering process which models
perceptual response to color differences (\Lab color space). The advantage of
the \Lab space filtering method over traditional RGB spaces is that the
former is a perceptually-uniform (metric) space, leading to a more consistent
definition of ``perceptually different'' colors. It is determined that the RGB
filtering method underestimates the perceived fractal dimension of lighter
colored patterns but not of darker ones, if the same selection criteria is
applied to each. Implications of the findings to Fechner's 'Principle of the
Aesthetic Middle' and Berlyne's work on perception of complexity are discussed.Comment: 21 pp LaTeX; two postscript figure
Quantum Iterated Function Systems
Iterated functions system (IFS) is defined by specifying a set of functions
in a classical phase space, which act randomly on an initial point. In an
analogous way, we define a quantum iterated functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include
Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies
Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk
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