Fractals: Geometry, Analysis and Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 November 2020) | Viewed by 10747

Special Issue Editor


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Guest Editor
Department of Mathematics, University of California, Riverside, CA 92521, USA
Interests: mathematical physics; functional and harmonic analysis; geometric analysis, partial differential equations (PDEs); dynamical systems; spectral geometry; fractal geometry; connections with number theory, geometry and spectral theory; arithmetic geometry and noncommutative geometry

Special Issue Information

Dear Colleagues,

The goal of this Special Issue is to publish a collection of interesting and novel mathematics or mathematical physics original research papers or research expository articles, on a broad variety of topics related to fractals, viewed either as geometric or analytic objects, and their applications. Only papers of sufficiently high quality to be published in good quality established scholarly mathematics, mathematical physics or applied mathematics journals will be considered for publication in this Special Issue.

Possible topics include, but are not limited to:

  • Analysis and partial differential equations on or off fractals.
  • Wave propagation in fractal media.
  • Vibrations of fractal drums.
  • Geometry (and/or geometric measure theory) of fractals.
  • Self-similar sets and their various extensions.
  • Dimension theory of fractals (including complex dimensions and fractal zeta functions).
  • Fractals and number theory.
  • Fractals and probability theory (e.g., random fractals, diffusions and other stochastic processes on fractals).
  • Fractals and dynamical systems.
  • Fractals and mathematical physics.
  • Fractals and computer graphics or scientific computation, experimental mathematics.
  • Nonsmooth analysis (e.g., metric measure spaces and doubling spaces).
  • Noncommutative fractal geometry; fractals and operator algebras.
  • Fractal tilings.
  • Quasicrystals (e.g., geometric description, mathematical theory of diffraction).

Keywords

  • Fractals and related topics
  • Geometry
  • Analysis
  • Mathematical physics
  • Dynamical systems
  • Number theory
  • Probability theory
  • Operator algebras
  • Scientific computation and computer graphics
  • Mathematical theory and applications

Published Papers (5 papers)

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19 pages, 4870 KiB  
Article
On Non-Tensor Product Bivariate Fractal Interpolation Surfaces on Rectangular Grids
by Vasileios Drakopoulos and Polychronis Manousopoulos
Mathematics 2020, 8(4), 525; https://doi.org/10.3390/math8040525 - 03 Apr 2020
Cited by 7 | Viewed by 2334
Abstract
Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal [...] Read more.
Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal interpolation surfaces is presented. Necessary conditions for the attractor of an iterated function system to be the graph of a continuous bivariable function which interpolates a given set of data are also presented here. Moreover, a comparative study for four of the most important constructions and attempts on rectangular grids is considered which points out some of their limitations and restrictions. Full article
(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)
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13 pages, 268 KiB  
Article
The Mean Minkowski Content of Homogeneous Random Fractals
by Martina Zähle
Mathematics 2020, 8(6), 883; https://doi.org/10.3390/math8060883 - 01 Jun 2020
Cited by 2 | Viewed by 1506
Abstract
Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski [...] Read more.
Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski dimension D is, in general, greater than its almost sure variant. Moreover, an integral representation extending that from the special deterministic case is derived. Full article
(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)
14 pages, 422 KiB  
Article
Non-Stationary Fractal Interpolation
by Peter Massopust
Mathematics 2019, 7(8), 666; https://doi.org/10.3390/math7080666 - 25 Jul 2019
Cited by 13 | Viewed by 2490
Abstract
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k N where each F k maps H ( X ) H ( X ) and arises [...] Read more.
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k N where each F k maps H ( X ) H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting. Full article
(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)
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18 pages, 363 KiB  
Article
Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions
by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane and Mohamad Maamoun Turkawi
Mathematics 2020, 8(7), 1179; https://doi.org/10.3390/math8071179 - 17 Jul 2020
Cited by 4 | Viewed by 1326
Abstract
We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) [...] Read more.
We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal. Full article
(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)
16 pages, 331 KiB  
Article
Periodic Intermediate β-Expansions of Pisot Numbers
by Blaine Quackenbush, Tony Samuel and Matt West
Mathematics 2020, 8(6), 903; https://doi.org/10.3390/math8060903 - 03 Jun 2020
Cited by 1 | Viewed by 2039
Abstract
The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β -shifts, namely transformations of the form [...] Read more.
The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β -shifts, namely transformations of the form T β , α : x β x + α mod 1 acting on [ α / ( β 1 ) , ( 1 α ) / ( β 1 ) ] , where ( β , α ) Δ is fixed and where Δ { ( β , α ) R 2 : β ( 1 , 2 ) and 0 α 2 β } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of ( β , α ) such that T β , α has the subshift of finite type property is dense in the parameter space Δ . Here, they proposed the following question. Given a fixed β ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of α in the fiber { β } × ( 0 , 2 β ) , so that T β , α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when α = 0 to the case when α ( 0 , 2 β ) . That is, we examine the structure of the set of eventually periodic points of T β , α when β is a Pisot number and when β is the n-th root of a Pisot number. Full article
(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)
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