- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 20:27:03
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JFG&vol=3&iss=1&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
3
2016
1
The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions
Pieter
Allaart
University of North Texas, DENTON, UNITED STATES
Continuous nowhere differentiable function, singular function, Cantor function, infinite derivative, ternary expansion, beta-expansion, Komornik–Loreti constant, Thue–Morse sequence, Hausdorff dimension
Okamoto's one-parameter family of self-affine functions $F_a: [0,1] \to [0,1]$, where $0 < a < 1$, includes the continuous nowhere differentiable functions of Perkins ($a=5/6$) and Bourbaki/Katsuura ($a=2/3$), as well as the Cantor function ($a=1/2$). The main purpose of this article is to characterize the set of points at which $F_a$ has an infinite derivative. We compute the Hausdorff dimension of this set for the case $a \leq 1/2$, and estimate it for $a > 1/2$. For all $a$, we determine the Hausdorff dimension of the sets of points where: (i) $F_a'=0$; and (ii) $F_a$ has neither a finite nor an infinite derivative. The upper and lower densities of the digit $1$ in the ternary expansion of $x \in [0,1]$ play an important role in the analysis, as does the theory of $\beta$-expansions of real numbers.
Real functions
Number theory
Measure and integration
1
31
10.4171/JFG/28
http://www.ems-ph.org/doi/10.4171/JFG/28
On the Hausdorff and packing measures of slices of dynamically defined sets
Ariel
Rapaport
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Self-similar set, self-affine set, Hausdorff measures, packing measures
Let $1 \le m < n$ be integers, and let $K \subset \mathbb{R}^{n}$ be a self-similar set satisfying the strong separation condition, and with dim $K = s > m$. We study the a.s. values of the $s-m$-dimensional Hausdorff and packing measures of $K \cap V$, where $V$ is a typical $n-m$-dimensional affine subspace. For $0
Measure and integration
33
74
10.4171/JFG/29
http://www.ems-ph.org/doi/10.4171/JFG/29
Magnetic fields on resistance spaces
Michael
Hinz
Universität Bielefeld, BIELEFELD, GERMANY
Luke
Rogers
University of Connecticut, STORRS, UNITED STATES
Resistance forms, Dirichlet forms, magnetic Laplacians, self-adjointness
On a metric measure space $X$ that supports a regular, strongly local resistance form we consider a magnetic energy form that corresponds to the magnetic Laplacian for a particle confined to $X$. We provide sufficient conditions for closability and self-adjointness in terms of geometric conditions on the reference measure without assuming energy dominance.
Measure and integration
Operator theory
75
93
10.4171/JFG/30
http://www.ems-ph.org/doi/10.4171/JFG/30