scholarly journals Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior

2021 ◽  
Vol 0 (0) ◽  
Silas L. Carvalho ◽  
Alexander Condori

Abstract In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q > 0 {q>0} , zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system ( X , T ) {(X,T)} (where X = M ℤ {X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q > 1 {q>1} , infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s ∈ ( 0 , 1 ) {s\in(0,1)} and each q > 1 {q>1} , zero lower s-generalized and infinite upper q-generalized dimensions.

Fractals ◽  
1996 ◽  
Vol 04 (03) ◽  
pp. 241-250 ◽  

Turbulent interface caused by the 2-dimensional Rayleigh-Taylor instability is investigated by direct numerical simulation. It is shown that the interface becomes fractal spontaneously in the case where there are initially multimode perturbations on the interface. The generalized dimensions and the singularity spectrum are obtained by applying the multifractal theory to the turbulent interface. The fractal dimension of the line interface is found to be 1.7–1.8, which is greater than that of turbulent/nonturbulent interface in a turbulent flow. Time evolution of the fractal dimensions of the interface is also investigated.

2019 ◽  
Vol 38 (7) ◽  
pp. 109-124
Arkan Mohammed

The non-geometric and irregular objects are considered as complex patterns. The geometric complexity is measured as space lling capacity by a factor known as a fractal dimension. Dierent techniques are proposed to nd this complexity measure according to the properties of the pattern. This paper is aimed to introduce a method for counting the dimension of the lled Julia fractal set generated by the Escape Time Algorithm using the method of spreading the points inside the proposed window. The resulted dimension is called Escape Time dimension. A new method to compute a correlation dimension of the Filled Julia fractal set is also proposed based on the Grassberger-Procaccia algorithm by computing the correlation function. A log-log graph of the correlation function versus the distances between every pair of points in the lled Julia fractal set is an approximation of the correlation dimension. Finally, a comparison between these two fractal dimensions of the led Julia fractal set which is generated by the Escape Time Algorithm is presented to show the efficiency of the proposed method.

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

2021 ◽  
Vol 11 (1) ◽  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

1999 ◽  
Vol 29 (9) ◽  
pp. 1301-1310 ◽  
Wojciech Borkowski

An application of fractal dimensions as measures of leaf complexity to morphometric studies and automated plant identification is presented. Detailed algorithms for the calculation of compass dimension and averaged mass dimension together with a simple method of grasping the scale range related variability are given. An analysis of complexity of more than 300 leaves from 10 tree species is reported. Several classical biometric descriptors as well as 16 fractal dimension features were computed on digitized leaf silhouettes. It is demonstrated that properly defined fractal dimension based features may be used to discriminate between species with more than 90% accuracy, especially when used together with other measures. It seems, therefore, that they can be utilized in computer identification systems and for purely taxonomical purposes.

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

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