Packing Dimension Research Articles

Some New Characterizations of Olsen’s Multifractal Functions

The aim of this work is to provide a representation of the multifractal Hausdorff and packing dimensions of compact sets in the Euclidean space in terms of the lower and upper local $$(q,\mu )$$ -dimensions of a probability measure on $$\mathbb {R}^n$$ , respectively. In addition, a relationship is established allowing to determine the Olsen’s multifractal functions by means of the corresponding measures’ versions. The multifractal functions are defined as the supremum over the lower and upper multifractal dimensions of all Borel probability measures.

Probabilistic Shaping in Time-Frequency-Packed Terabit Superchannel Transmission

To combine the individual benefits of probabilistic-shaping (PS) and time-frequency-packing (TFP), we consider for the first time PS-TFP wavelength-division multiplexing (WDM) superchannels. However, TFP introduces inter-symbol interference (ISI) and/or inter-carrier interference (ICI). Moreover, the presence of reconfigurable optical add-drop multiplexers in the fiber links may further degrade the system performance. In this letter, we efficiently handle such challenges to present PS-TFP superchannels enabling Terabit-per-second data rates. For this, we perform a joint ISI and ICI channel estimation in tandem with turbo equalization to mitigate the interference. We investigate optimizing the parameters in the shaping and packing dimensions to achieve a desired target spectral efficiency. We show through our numerical results that such an optimized PS-TFP transmission leads up to 1.2 dB performance improvement over an unshaped Nyquist WDM system under similar conditions.

Dimension of Gibbs measures with infinite entropy

We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an almost sure value for the symbolic dimension. We also show that the lower local dimension dimension is almost surely equal to zero, while the upper local dimension is almost surely equal to the symbolic dimension. In particular, we prove that a large class of Gibbs measures with infinite entropy for the Gauss map have Hausdorff dimension zero and packing dimension equal to 1/2, and so such measures are not exact dimensional.

An Expanded Batch-to-Batch Correction for IAPSO Standard Seawater

AbstractWe expanded the batch-to-batch offsets of The International Association for the Physical Sciences of the Oceans (IAPSO) Standard Seawater (SSW) batches P145–P163 by intercomparison measurements using salinometers. On the basis of our results, we recommend using the correction factors instead of the offsets to correct the batch-to-batch differences, especially for salinity data outside the range of 30–40 g kg−1. We evaluated the expanded batch-to-batch correction factors by applying them to time series salinity data collected in the northwestern North Pacific Ocean and found that they are effective for detecting recent freshening (−0.6 ± 0.1 × 10−3 g kg−1 decade−1) in the deep North Pacific, which might be related to a reduction of the formation rate of Antarctic Bottom Water. We also evaluated the SSW linearity pack by applying the batch-to-batch correction factors. Linearity errors of the salinometers estimated from decade resistance substituters were consistent with the results of the linearity pack measurements. To correct the linearity errors of a salinometer, it might be suitable to use the more detailed distribution of those estimated from the decade resistance substituter than the linearity pack measurements. Since the cause of large batch-to-batch differences is still unclear, a reference seawater that is more robust and stable than SSW might be necessary to establish a high-level of international comparability of salinity measurements; the Multiparametric Standard Seawater (MSSW) currently under development might be a candidate for such reference seawater, because MSSW is expected to be more stable than SSW not only in practical salinity but also in absolute salinity.

Hausdorff and packing dimensions of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.

Packing dimensions of basins in the space of sequences

We consider a space of infinite signals composed of finite-alphabet letters. Each signal generates a sequence of empirical measures on the alphabet and a limit set corresponding to this sequence. The space of signals is partitioned into narrow basins consisting of signals with identical limit sets for the empirical measures, and the packing dimension is computed for each narrow basin.

On a linearity between fractal dimension and order of fractional calculus in Hölder space

Abstract In this paper, the linear relationship between fractal dimensions and the order of fractional calculus of functions in Holder space has been mainly investigated. Under specific Holder condition, the linear connection between Box dimension and the order of Riemann-Liouville fractional integral and derivative has been proved. This linear connection is also established with K-dimension and Packing dimension. Some function examples have been given in the end.

On the Assouad dimension of projections

Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} \pi_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This is a strong variant of Marstrand's projection theorem for Assouad dimension, whose analogue is not true for other common notions of fractal dimension, such as Hausdorff or packing dimension.

Attainable values for the Assouad dimensionof projections

We prove that for an arbitrary upper semi-continuous function $\phi\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} \pi F = \phi(\pi)$ for all $\pi \in G(1,2)$, where $\pi F$ is the orthogonal projection of $F$ onto the line $\pi$. In particular, this shows that the Assouad dimension of orthogonal projections can take on any finite or countable number of distinct values on a set of projections with positive measure. It was previously known that two distinct values could be achieved with positive measure. Recall that for other standard notions of dimension, such as the Hausdorff, packing, upper or lower box dimension, a single value occurs almost surely.

Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings

We study infinite iterated function systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focussing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

Shrinking targets and eventually always hitting points for interval maps

We study shrinking target problems and the set of eventually always hitting points. These are the points whose first n iterates will never have empty intersection with the nth target for sufficiently large n. We derive necessary and sufficient conditions on the shrinking rate of the targets for to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville–Pomeau map. We also obtain results for the Gauß map and correspondingly for the maximal digits in continued fraction expansions. In the case of -transformations we also compute the packing dimension of complementing already known results on the Hausdorff dimension of .

On the Multifractal Analysis of Branching Random Walk on Galton–Watson Tree with Random Metric

We consider a branching random walk $$S_nX(t)$$ on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set $$E(\alpha )$$ of infinite branches in the boundary of tree endowed with random metric along which the average of $$S_n X(t)/n$$ have a given limit point.

Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions

ABSTRACTWe consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.

On the mutual singularity of multifractal measures

<p style='text-indent:20px;'>The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.</p>

Dimension of the Non-Differentiability Subset of the Cantor Function

The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Packing dimension of Tμλ is log2/log3. The log2 = loge2 is that if ax = N (a >0, and a≠1), then the number x is called the logarithm of N with a base, recorded as x = logaN, read as the logarithm of N with a base, where a is called logarithm Base number, N is called true number.

On the exact dimension of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of the argument used by Kahane 1987 on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J . As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets Eα of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

An improved bound on the packing dimension of Furstenberg sets in the plane

Let 0 \leq s \leq 1 . A set K \subset \mathbb{R}^{2} is a Furstenberg s -set if for every unit vector e \in S^{1} , some line L_{e} parallel to e satisfies \dim_H [K \cap L_{e}] \geq s. The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg s -sets. Wolff proved that \dim_H K \geq \max\{s + 1/2,2s\} and conjectured that \dim_H K \geq (1 + 3s)/2 . The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that \dim_H K \geq 1 + \epsilon for Furstenberg 1/2 -sets K , where \epsilon > 0 is an absolute constant. In the present paper, I prove a similar \epsilon -improvement for all 1/2 < s < 1 , but only for packing dimension: \dim_p K \geq 2s + \epsilon for all Furstenberg s -sets K \subset \mathbb{R}^{2} , where \epsilon > 0 only depends on s . The proof rests on a new incidence theorem for finite collections of planar points and tubes of width \delta > 0 . As another corollary of this theorem, I obtain a small improvement for Kaufman's estimate from 1968 on the dimension of exceptional sets of orthogonal projections. Namely, I prove that if K \subset \mathbb{R}^{2} is a linearly measurable set with positive length, and 1/2 < s < 1 , then \dim_H \{e \in S^{1} : \dim_p \pi_{e}(K) \leq s\} \leq s - \epsilon for some \epsilon > 0 depending only on s . Here \pi_{e} is the orthogonal projection onto the line spanned by e .

Multifractal spectra of Moran measures without local dimension

A measure without local dimension is a measure such that local dimension does not exist for any point in its support. In this paper, we construct such a class of Moran measures and study their lower and upper local dimensions. We show that the related ‘free energy’ function (Lq-spectrum) does not exist. Nevertheless, we can obtain the full Hausdroff and packing dimension spectra for level sets defined by lower and upper local dimensions. They can be viewed as a generalized multifractal formalism.

Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric L\'evy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for $\alpha$-stable L\'evy processes with $1<\alpha<2$. Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

A Multifractal Formalism for Hewitt–Stromberg Measures

In the present work, we give a new {\it multifractal formalism} for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt-Stromberg measures and that this formalism is completely parallel to Olsen's multifractal formalism which based on the Hausdorff and packing measures.