Journal ArticleDOI
An equivalent definition of packing dimension and its application
TLDR
In this article, an equivalent definition of packing dimension is given for a set in d-dimensional Euclidean space by using its component sets as packings, applied to determine the packing dimensions of a class of subsets with prescribed relative group frequencies.Abstract:
An equivalent definition of packing dimension is given for a set in d-dimensional Euclidean space by using its component sets as packings. It is applied to determine the packing dimensions of a class of subsets with prescribed relative group frequencies.read more
Citations
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Journal ArticleDOI
Pointwise dimensions of general Moran measures with open set condition
Jinjun Li,Jinjun Li,Min Wu +2 more
TL;DR: In this paper, the authors obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in Ωd under the strong separation condition and proved that the result is still true under the open set condition.
Journal ArticleDOI
The Effects of the Riemann-Liouville Fractional Integral on the Box Dimension of Fractal Graphs of HÖLDER Continuous Functions
TL;DR: In this article, the linearity of the dimensional decrease effect of Riemann-Liouville fractional integral is explored and it is proved that if the Box dimension of the graph of an α-Holder c...
Journal ArticleDOI
Fractal stokes’ theorem based on integrals on fractal manifolds
Junru Wu,Chengyuan Wang +1 more
TL;DR: In this paper, the Hausdorff integral on fractal sets with one or lower dimension was introduced via measure theory, and the definition of the integral on a fractal set was given.
Journal ArticleDOI
The parameter distribution set for a self-similar measure
TL;DR: In this paper, the lower distribution set corresponds to the cylindrical lower or upper local dimension set for a self-similar measure on a selfsimilar set satisfying the open set condition.
Journal ArticleDOI
The local dimensions of some Moran measures with open set condition
TL;DR: In this article, Dai and Liu obtained the formula of local dimensions of some Moran measures on Moran sets in R d under the strong separation condition and proved that the result is still true under the open set condition.
References
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Book
Fractal Geometry: Mathematical Foundations and Applications
TL;DR: In this article, a mathematical background of Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractality products of fractal intersections of fractalities.
Book
An Introduction to Ergodic Theory
TL;DR: The first part of the text as discussed by the authors provides an introduction to ergodic theory suitable for readers knowing basic measure theory, including recurrence properties, mixing properties, the Birkhoff Ergodic theorem, isomorphism, and entropy theory.
Book ChapterDOI
An Introduction to Ergodic Theory
TL;DR: Ergodic theory concerns with the study of the long-time behavior of a dynamical system as mentioned in this paper, and it is known as Birkhoff's ergodic theorem, which states that the time average exists and is equal to the space average.
Book
Techniques in fractal geometry
TL;DR: A review of fractal geometry can be found in this article, with a focus on the Ergodic Theorem and Fractals, as well as the renewalal theorem and fractals.
Journal ArticleDOI
Multifractal decompositions of Moran fractals
Robert Cawley,R. Daniel Mauldin +1 more
TL;DR: In this article, a rigorous construction and generalization of the multifractal decomposition for Moran fractals with infinite product measure is presented, which is specified by a system of nonnegative weights in the partition sum.