Narrow Results

In this paper, we estimate the oscillation of harmonic functions on the $(N-1)$-dimensional Sierpiński gasket $K^N$, where $K^3$ is the classical Sierpiński gasket. By using this result, we prove that both the box dimension and the Hausdorff dimension of graphs of harmonic functions on $K^N$ are ${log}_{2}N$.

Denote by $E_{{ℱ}}$ and $E_{{ℐ}}$ the sets of real numbers whose Engel expansion digits contain arbitrarily long arithmetic progressions and arbitrarily long arithmetic progressions with arbitrary common difference, respectively. In this paper, the sizes of $E_{{ℱ}}$ and $E_{{ℐ}}$ are investigated from the measure-theoretical, fractal and topological viewpoints. We prove that $E_{{ℱ}}$ and $E_{{ℐ}}$ are of Lebesgue measure zero but have full Hausdorff dimension. Moreover, $E_{{ℐ}}$ is of first category while $E_{{ℱ}}$ is residual.

Measures were constructed on symbolic spaces that satisfy an extended multifractal formalism, where Olsen’s functions $b$ and $B$ differ, and their Legendre transforms have the expected interpretation in terms of dimensions. These measures were composed with a Gray code and projected onto the unit interval to obtain doubling measures. It was demonstrated that the projected measure retains the same Olsen’s functions as the original and also satisfies the extended multifractal formalism. In this paper, we show that the use of a Gray code is not essential to achieve these results, even when dealing with non-doubling measures. Moreover, general results on multifractal analysis of inhomogeneous multinomial measures with their non-doubling projections are obtained. The key points of the proof include two main components: the study of weak doubling properties and the method of constructing auxiliary measure to get sharp bound for the dimension under consideration.

In this paper, we determine the Hausdorff dimension of the limsup and liminf sets where the Pierce expansion digits grow at a rate no slower than any positive function defined on the natural numbers. Additionally, we provide similar analyses concerning the gaps and ratios between consecutive digits.

We use means of formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry.

Our results are used to provide a series of simple examples for the non-coincidence of limit sets and attractors for infinite iterated function systems.

Decay of various quantities (return or survival probability, correlation functions) in time are the basis of a multitude of important and interesting phenomena in quantum physics, ranging from spectral properties, resonances, return and approach to equilibrium, to dynamical stability properties and irreversibility and the "arrow of time" in [Asymptotic Time Decay in Quantum Physics (World Scientific, 2013)].

In this review, we study several types of decay — decay in the average, decay in the L^p-sense, and pointwise decay — of the Fourier–Stieltjes transform of a measure, usually identified with the spectral measure, which appear naturally in different mathematical and physical settings. In particular, decay in the L^p-sense is related both to pointwise decay and to decay in the average and, from a physical standpoint, relates to a rigorous form of the time-energy uncertainty relation. Both decay on the average and in the L^p-sense are related to spectral properties, in particular, absolute continuity of the spectral measure. The study of pointwise decay for singular continuous measures (Rajchman measures) provides a bridge between ergodic theory, number theory and analysis, including the method of stationary phase. The theory is illustrated by some new results in the theory of sparse models.

We consider a class of subsets of the general Sierpinski carpet which is characterized by insisting that the allowed digits in the expansion occur with prescribed mixing group frequencies, determine their Hausdorff dimensions and give the necessary and sufficient conditions for their corresponding Hausdorff measures to be positive and finite.

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on ${\mathbb{P}}^1$. Results on the iterative dynamics of such a correspondence can now be applied to the study of the rational semigroup. We focus on an invariant measure for the aforementioned correspondence — known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyze the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup.

Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space–time. We show that the Hausdorff dimension depends on the deformation parameter $a$ and the resolution $\mathrm{\Delta }x$ for both nonrelativistic and relativistic quantum particle. For the nonrelativistic case, it is seen that Hausdorff dimension is always less than 2 in the noncommutative space–time. For relativistic quantum particle, we find the Hausdorff dimension increases with the noncommutative parameter, in contrast to the commutative space–time. We show that noncommutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space–time. By imposing self-similarity condition on the path of nonrelativistic and relativistic quantum particle in noncommutative space–time, we derive the corresponding generalized uncertainty relation.

We mainly investigate the likely limit sets and the kneading sequences of unimodal Feigenbaum's maps (Feigenbaum's map can be regarded as the fixed point of the renormalization operator , where λ is to be determined). First, we estimate the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum's map and then for every decimal s ∈ (0, 1), we construct a unimodal Feigenbaum's map which has a likely limit set with Hausdorff dimension s. Second, we prove that the kneading sequences of unimodal Feigenbaum's maps are uniformly almost periodic points of the shift map but not periodic ones.

In this article, we investigate the energy cascade of three-dimensional turbulent flows, in which the break-up process of eddy is quasi-self-similar. Mathematically this kind of turbulence with quasi-self-similar structure eddies can be regarded as cookie-cutter system, and can be generated by self-similar iterated function system (IFS) with added nonlinear disturbance. Using Bowen's result, we can calculate the exponent of dissipative correlated function, dissipated velocity, energy spectrum supported on cookie-cutter system. The present results show that the β-model is feasible for this kind of quasi-self-similar turbulence.

This is an analytical analysis of a previously published research for a percolation simulation. In that research the effect of mutations on adaptability was investigated in a bit-string model of invading species in a random environment. However, analytical analysis was missing which will be the topic here. The Hausdorff dimensions are calculated for the fractals and the conditions on invasion are analyzed analytically by manipulation of partial differential equations. Thus, various conclusions may be reached without having to run long simulations.

The challenges in productivity of satellite mobile devices are growing rapidly to overcome the question of miniaturization. The intention is to supply the electrical and microwave properties of materials by discovering their outstanding electronic properties. Neodymium Zinc Titanate (NZT) can be a promising ferroelectric material due to its stable dielectric and microwave properties. The grain size and shape of NZT have a strong influence on overall material performances. Therefore, shape, reconstruction and property of the coming compound take an important part and can be predicted before being utilized in the devices. The significant of this research is to define ferroelectric properties of NZT and to characterize it by using Fractal Nature Analysis (FNA). FNA is a powerful mathematical technique that could be applied to improve the grain shape and interface reconstruction. The fractal structure is identified by its self-similarity. The self-similarity of an object means a repetition of shapes in smaller scales. A measure of this structure is computed using the Hausdorff dimension. It is for the first time in this investigation the Fractal analysis method is applied for the microwave materials microstructure reconstruction which makes this research an innovative work and will open the door for Curie–Weiss law fractal correction. In connection to our previous research for dielectric properties fractalization, we had some characterization and reconstruction data which include the Hausdorff dimension (HD).

A new class of fractals called hyperfractals is systematically investigated. Their underlying fractals in supporting spaces are multivalued fractals developed by ourselves in our earlier papers [Andres & Fiser, 2004; Andres et al., 2005]. In the case of uniqueness, they also coincide with the supports of invariant measures of the associated Markov–Feller hyperoperators. We prove their existence by means of fixed point theorems in hyperspaces and calculate, in particular cases of generating hypermaps, their Hausdorff dimension. Two illustrative examples are supplied.

In this paper, we consider the well-posedness and asymptotic behaviors of solutions of the fractional complex Ginzburg–Landau equation with the initial and periodic boundary conditions in two spatial dimensions. We explore the existence and uniqueness of global smooth solution by means of the Galerkin method and establish the existence of the global attractor. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor are presented.

In this paper, we prove the existence of pullback and uniform attractors for a nonautonomous Liénard equation. The relation among these attractors is also discussed. After that, we consider that the Liénard equation includes forcing terms which belong to a class of functions extending periodic and almost periodic functions recently introduced by Kloeden and Rodrigues [2011]. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate these results with a numerical simulation: we present a simulation showing the pullback attractor for the nonautonomous Van der Pol equation, an important special case of the nonautonomous Liénard equation.

New classes of Lyapunov-type functions are suggested for the estimation of Lyapunov dimension of Lorenz-like systems. Lyapunov dimension formulas for Lorenz and Tigan systems are obtained.

We introduce the k-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contracting. We analyze the limit spaces of a particular family of contracting Hanoi groups, , and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.

We show that the Hausdorff dimension of the closure of the second Grigorchuk group is $43/128$. Furthermore, we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normalizer in the full automorphism group of the tree.