- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 19:15:40
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JFG&vol=2&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
2
2015
3
Higher moments for random multiplicative measures
Kenneth
Falconer
University of St Andrews, ST ANDREWS, GREAT BRITAIN
Multiplicative measures, cascades, moments, random fractals
We obtain a condition for the $L^q$-convergence of martingales generated by random multiplicative cascade measures for $q>1$ without any self-similarity requirements on the cascades.
Probability theory and stochastic processes
Measure and integration
229
247
10.4171/JFG/20
http://www.ems-ph.org/doi/10.4171/JFG/20
On McMullen-like mappings
Antonio
Garijo
Universitat Rovira i Virgili, TARRAGONA, SPAIN
Sébastien
Godillon
University de Barcelona, BARCELONA, SPAIN
Complex dynamics, Julia sets, rational maps, McMullen family
We introduce a generalization of particular dynamical behavior for rational maps. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}(z)=z^n+\lambda/z^d$ for $|\lambda|\neq 0$ small enough is a Cantor set of circles if and only if $1/n+1/d
Dynamical systems and ergodic theory
Functions of a complex variable
249
279
10.4171/JFG/21
http://www.ems-ph.org/doi/10.4171/JFG/21
Connectedness locus for pairs of affine maps and zeros of power series
Boris
Solomyak
Bar-Ilan University, RAMAT GAN, ISRAEL
Self-affine sets, connectedness locus, zeros of power series
We study the connectedness locus $\mathcal N$ for the family of iterated function systems of pairs of a ffine-linear maps in the plane (the non-self-similar case). First results on the set $\mathcal N$ were obtained in joint work with P. Shmerkin [11]. Here we establish rigorous bounds for the set $\mathcal N$ based on the study of power series of special form. We also derive some bounds for the region of “$\ast$-transversality” which have applications to the computation of Hausdorff measure of the self-affi ne attractor. We prove that a large portion of the set $\mathcal N$ is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set $\mathcal N$ has many zero angle “cusp corners,” at certain points with algebraic coordinates.
Measure and integration
Functions of a complex variable
281
308
10.4171/JFG/22
http://www.ems-ph.org/doi/10.4171/JFG/22
On the Fourier dimension and a modification
Fredrik
Ekström
Lund University, LUND, SWEDEN
Tomas
Persson
Lund University, LUND, SWEDEN
Jörg
Schmeling
Lund University, LUND, SWEDEN
Stability of Fourier dimension, modified Fourier dimension
We give a su fficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the de finition of the Fourier dimension so that it becomes countably stable, and show that for each $s$ there is a class of sets such that a measure has modi ed Fourier dimension greater than or equal to s if and only if it annihilates all sets in the class.
Fourier analysis
Measure and integration
309
337
10.4171/JFG/23
http://www.ems-ph.org/doi/10.4171/JFG/23