- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 17:43:21
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JFG&vol=1&iss=3&update_since=2024-03-28
Journal of Fractal Geometry
J. Fractal Geom.
JFG
2308-1309
2308-1317
Measure and integration
General
10.4171/JFG
http://www.ems-ph.org/doi/10.4171/JFG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2014
3
A topological separation condition for fractal attractors
Tim
Bedford
University of Strathclyde, Glasgow, GREAT BRITAIN
Sergiy
Borodachov
Towson University, TOWSON, UNITED STATES
Jeffrey
Geronimo
Georgia Institute of Technology, ATLANTA, UNITED STATES
Minimality property, separation condition, Hausdorff dimension, similarity dimension, open set condition, Markov partition property, self-similar sets
We consider finite systems of contractive homeomorphisms of a complete metric space, which satisfy the minimality property. In general this separation condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We prove that this separation condition is equivalent to the strong Markov property (see definition below). We also show that the set of $N$-tuples of contractive homeomorphisms, having the minimality property, is a $G_\delta$ set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings in the sequence be strictly less than one. We find a class of $N$-tuples of $d\times d$ invertible contraction matrices, which define systems of affine mappings in $\mathbb R^d$ having the minimality property for almost every $N$-tuple of fixed points with respect to the $Nd$-dimensional Lebesgue measure.
Measure and integration
Dynamical systems and ergodic theory
243
271
10.4171/JFG/7
http://www.ems-ph.org/doi/10.4171/JFG/7
Dimension of slices of Sierpiński-like carpets
Balázs
Bárány
Polish Academy of Sciences, WARSAW, POLAND
Michał
Rams
Polish Academy of Sciences, Warsaw, POLAND
Sierpiński carpets, projection
We investigate the dimension of intersections of the Sierpiński-like carpets with lines. We show a sufficient condition that for a fixed rational slope the dimension of almost every intersection with respect to the natural measure is strictly greater than $s–1$, and almost every intersection with respect to the Lebesgue measure is strictly less than $s–1$, where s is the Hausdorff dimension of the carpet. Moreover, we give partial multifractal spectra for the Hausdorff and packing dimension of slices.
Measure and integration
273
294
10.4171/JFG/8
http://www.ems-ph.org/doi/10.4171/JFG/8
Measures and functions with prescribed homogeneous multifractal spectrum
Zoltán
Buzcolich
Eötvös University, BUDAPEST, HUNGARY
Stéphane
Seuret
Université Paris-Est Créteil, CRETEIL, FRANCE
Hausdorff dimensions and measures, multifractal analysis, Hölder exponent, regularity of measures and local dimensions, wavelets
In this paper we construct measures supported in $\zu$ with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of $\zu$ has the same multifractal spectrum as the whole measure. The spectra $f$ that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of $\zu$ and satisfy $f(h)\leq h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure, that we call Darboux theorem for multifractal spectra of measures: the support of its spectrum within $[0,1]$ must be an interval. This result is optimal, because there exists a HM measure with spectrum supported by $\zu\cup \{ 2 \}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.
Real functions
Measure and integration
Fourier analysis
295
333
10.4171/JFG/9
http://www.ems-ph.org/doi/10.4171/JFG/9
The shape of the dust-likeness locus of self-similar sets
Miwa
Aoki
Nara Women's University, NARA, JAPAN
Masayo
Fujimura
National Defense Academy of Japan, YOKOSUKA, KANAGAWA, JAPAN
Masahiko
Taniguchi
Nara Women's University, NARA, JAPAN
Self-similar set, (contractive linear) IFS, dust-likeness locus
We prove that the dust-likeness locus in the deformation space of a contractive holomorphic linear iterated function systemis coincidentwith the quasiconformal deformation space. Also, we determine explicitly the dust-likeness locus restricted to several slices which are different from the Mandelbrot one, and provide specific examples that show diversity of self-similar sets.
Measure and integration
Dynamical systems and ergodic theory
335
347
10.4171/JFG/10
http://www.ems-ph.org/doi/10.4171/JFG/10