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Статті в журналах з теми "Dimension de Hausdorff"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 4 травня 2024

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1

Myjak, Józef. "Some typical properties of dimensions of sets and measures." Abstract and Applied Analysis 2005, no. 3 (2005): 239–54. http://dx.doi.org/10.1155/aaa.2005.239.

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Анотація:

This paper contains a review of recent results concerning typical properties of dimensions of sets and dimensions of measures. In particular, we are interested in the Hausdorff dimension, box dimension, and packing dimension of sets and in the Hausdorff dimension, box dimension, correlation dimension, concentration dimension, and local dimension of measures.

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2

Conidis, Chris J. "A real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one." Journal of Symbolic Logic 77, no. 2 (June 2012): 447–74. http://dx.doi.org/10.2178/jsl/1333566632.

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Анотація:

AbstractRecently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

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3

Leonov, G. "Hausdorff–Lebesgue Dimension of Attractors." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1750164. http://dx.doi.org/10.1142/s0218127417501644.

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Анотація:

Definitions of Hausdorff–Lebesgue measure and dimension are introduced. Combination of Hausdorff and Lebesgue ideas are used. Methods for upper and lower estimations of attractor dimensions are developed.

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4

DAS, TUSHAR, LIOR FISHMAN, DAVID SIMMONS, and MARIUSZ URBAŃSKI. "Badly approximable points on self-affine sponges and the lower Assouad dimension." Ergodic Theory and Dynamical Systems 39, no. 3 (June 20, 2017): 638–57. http://dx.doi.org/10.1017/etds.2017.42.

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Анотація:

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

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5

ZÄHLE, M. "THE AVERAGE FRACTAL DIMENSION AND PROJECTIONS OF MEASURES AND SETS IN Rn." Fractals 03, no. 04 (December 1995): 747–54. http://dx.doi.org/10.1142/s0218348x95000667.

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In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.

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6

Urbański, Mariusz. "Transfinite Hausdorff dimension." Topology and its Applications 156, no. 17 (November 2009): 2762–71. http://dx.doi.org/10.1016/j.topol.2009.01.025.

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7

J�rgensen, H., and L. Staiger. "Local Hausdorff dimension." Acta Informatica 32, no. 5 (August 1, 1995): 491–507. http://dx.doi.org/10.1007/s002360050025.

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8

Jürgensen, H., and L. Staiger. "Local Hausdorff dimension." Acta Informatica 32, no. 5 (May 1995): 491–507. http://dx.doi.org/10.1007/bf01213081.

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9

SIMMONS, DAVID. "A Hausdorff measure version of the Jarník–Schmidt theorem in Diophantine approximation." Mathematical Proceedings of the Cambridge Philosophical Society 164, no. 3 (April 5, 2017): 413–59. http://dx.doi.org/10.1017/s0305004117000214.

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Анотація:

AbstractWe solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalising to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorfff-measure of the set of matrices which are not ψ-approximable, given a dimension functionfand a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorfff-measure of the set of matrices which are ψ-approximable.

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10

SHANG, LEI, and MIN WU. "SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS." Fractals 28, no. 03 (May 2020): 2050047. http://dx.doi.org/10.1142/s0218348x20500474.

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Анотація:

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].

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11

Fernández-Martínez, Manuel, Juan Luis García García Guirao, and Miguel Ángel Sánchez-Granero. "Calculating Hausdorff Dimension in Higher Dimensional Spaces." Symmetry 11, no. 4 (April 18, 2019): 564. http://dx.doi.org/10.3390/sym11040564.

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Анотація:

In this paper, we prove the identity dim H(F) = d dim H(a?1(F)), where dim H denotesHausdorff dimension, F Rd, and a : [0, 1] ! [0, 1]d is a function whose constructive definition isaddressed from the viewpoint of the powerful concept of a fractal structure. Such a result standsparticularly from some other results stated in a more general setting. Thus, Hausdorff dimension ofhigher dimensional subsets can be calculated from Hausdorff dimension of 1-dimensional subsets of[0, 1]. As a consequence, Hausdorff dimension becomes available to deal with the effective calculationof the fractal dimension in applications by applying a procedure contributed by the authors inprevious works. It is also worth pointing out that our results generalize both Skubalska-Rafajłowiczand García-Mora-Redtwitz theorems.

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12

Soltanifar, Mohsen. "The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals." Mathematics 10, no. 5 (February 24, 2022): 706. http://dx.doi.org/10.3390/math10050706.

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Анотація:

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces.

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13

Attia, Najmeddine, and Bilel Selmi. "On the Fractal Measures and Dimensions of Image Measures on a Class of Moran Sets." Mathematics 11, no. 6 (March 21, 2023): 1519. http://dx.doi.org/10.3390/math11061519.

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Анотація:

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ-invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures.

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14

Falconer, Kenneth J., Jonathan M. Fraser, and Tom Kempton. "Intermediate dimensions." Mathematische Zeitschrift 296, no. 1-2 (December 26, 2019): 813–30. http://dx.doi.org/10.1007/s00209-019-02452-0.

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Анотація:

AbstractWe introduce a continuum of dimensions which are ‘intermediate’ between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $$|U| \le |V|^\theta $$ | U | ≤ | V | θ for all sets U, V used in a particular cover, where $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] is a parameter. Thus, when $$\theta =1$$ θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when $$\theta =0$$ θ = 0 there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of $$\theta $$ θ ), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman’s lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford–McMullen carpets.

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15

WANG, JUN, and KUI YAO. "DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION." Fractals 25, no. 01 (February 2017): 1730001. http://dx.doi.org/10.1142/s0218348x1730001x.

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Анотація:

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

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16

Peres, Yuval. "The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure." Mathematical Proceedings of the Cambridge Philosophical Society 116, no. 3 (November 1994): 513–26. http://dx.doi.org/10.1017/s0305004100072789.

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AbstractWe show that the self-affine sets considered by McMullen in [11] and by Bedford in [1] have infinite Hausdorff measure in their dimension, except in the (rare) cases where the Hausdorff dimension coincides with the Minkowski (≡ box) dimension. More precisely, the Hausdorff measure of such a self-affine set K is infinite in the gauge(where γ is the Hausdorff dimension of K, and c > 0 is small). The Hausdorff measure of K becomes zero if 2 is replaced by any smaller number in the formula for the gauge ø.

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17

YAYAMA, YUKI. "Dimensions of compact invariant sets of some expanding maps." Ergodic Theory and Dynamical Systems 29, no. 1 (February 2009): 281–315. http://dx.doi.org/10.1017/s014338570800014x.

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AbstractWe study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a ‘general Sierpiński carpet’ was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpiński carpet represented by a shift of finite type. We give some conditions under which a general Sierpiński carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

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18

Körner, Thomas William. "Hausdorff and Fourier dimension." Studia Mathematica 206, no. 1 (2011): 37–50. http://dx.doi.org/10.4064/sm206-1-3.

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19

MA, DONGKUI, and MIN WU. "ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY." Fractals 18, no. 03 (September 2010): 363–70. http://dx.doi.org/10.1142/s0218348x10004956.

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Анотація:

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

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20

Soltanifar, Mohsen. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals." Mathematics 9, no. 13 (July 1, 2021): 1546. http://dx.doi.org/10.3390/math9131546.

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How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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21

Dudko, Artem, Igors Gorbovickis, and Warwick Tucker. "Lower bounds on the Hausdorff dimension of some Julia sets." Nonlinearity 36, no. 5 (April 11, 2023): 2867–93. http://dx.doi.org/10.1088/1361-6544/acc71b.

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Abstract We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial p F e i g ( z ) = z 2 + c F e i g . In addition to that, we construct a piecewise constant function on [ − 2 , 2 ] that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials p c ( z ) = z 2 + c with c ∈ [ − 2 , 2 ] . Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial p c ( z ) = z 2 + c , is a C 1-smooth function of the real parameter c on the interval c ∈ ( c F e i g , − 3 / 4 ) .

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22

CHUNG, YONG MOO. "BIRKHOFF SPECTRA FOR ONE-DIMENSIONAL MAPS WITH SOME HYPERBOLICITY." Stochastics and Dynamics 10, no. 01 (March 2010): 53–75. http://dx.doi.org/10.1142/s021949371000284x.

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Анотація:

We study the multifractal analysis for smooth dynamical systems in dimension one. It is given a characterization of the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.

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23

Mařík, Jan. "The Hausdorff dimension of some special plane sets." Mathematica Bohemica 119, no. 4 (1994): 359–66. http://dx.doi.org/10.21136/mb.1994.126119.

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24

Cao, Ziang. "Several Methods for Calculating and Estimating the Hausdorff Dimension." Frontiers in Computing and Intelligent Systems 5, no. 1 (September 12, 2023): 28–32. http://dx.doi.org/10.54097/fcis.v5i1.11588.

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Анотація:

Among various fractal dimensions, the Hausdorff dimension is the most widely used and fundamental one. However, in many cases, calculating or estimating its value can be challenging. This article systematically introduces several methods for calculating the Hausdorff dimension, starting with its definition and properties. In method one, the dimension is estimated using upper and lower bounds. For the upper bound, it usually requires finding a specific covering, while for the lower bound, the mass distribution principle is commonly employed for estimation; In method two, the clever use of potential theory is applied to estimate the Hausdorff dimension; In method three, for some high-dimensional cases, the dimension can be reduced using the projection theorem, and then the original dimension is estimated by computing the dimension of the lower-dimensional cases; In method four, the dimension estimation of the Cartesian product of two sets is considered, and several theorems are used to provide upper and lower bounds for the dimension; In method five, the focus is on the dimension calculation of self-similar sets. For such sets, under the condition of open sets, heuristic methods can be used for calculation; In method six, analogous to self-similar sets, the estimation of the dimension for self-affine sets is summarized.

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25

STALLARD, GWYNETH M. "DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS." Bulletin of the London Mathematical Society 33, no. 6 (November 2001): 689–94. http://dx.doi.org/10.1112/s0024609301008426.

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It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

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26

Kern, Peter, and Ercan Sönmez. "On the carrying dimension of occupation measures for self-affine random fields." Probability and Mathematical Statistics 39, no. 2 (December 19, 2019): 459–79. http://dx.doi.org/10.19195/0208-4147.39.2.12.

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Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle 1984, 1988, 1990, 1991. Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.

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27

Bedford, Tim. "On Weierstrass-like functions and random recurrent sets." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 2 (September 1989): 325–42. http://dx.doi.org/10.1017/s0305004100078142.

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Анотація:

AbstractA construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

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28

FRIEDLAND, SHMUEL. "Discrete Lyapunov exponents and Hausdorff dimension." Ergodic Theory and Dynamical Systems 20, no. 1 (February 2000): 145–72. http://dx.doi.org/10.1017/s0143385700000080.

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Анотація:

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

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29

LIAO, LINGMIN, JIHUA MA, and BAOWEI WANG. "Dimension of some non-normal continued fraction sets." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 1 (July 2008): 215–25. http://dx.doi.org/10.1017/s0305004108001291.

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AbstractWe consider certain sets of non-normal continued fractions for which the asymptotic frequencies of digit strings oscillate in one or other ways. The Hausdorff dimensions of these sets are shown to be the same value 1/2 as long as they are non-empty. An interesting example among them is the set of “extremely non-normal continued fractions” which was previously conjectured to be of Hausdorff dimension 0.

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30

Lempp, Steffen, Joseph S. Miller, Keng Meng Ng, Daniel D. Turetsky, and Rebecca Weber. "Lowness for effective Hausdorff dimension." Journal of Mathematical Logic 14, no. 02 (December 2014): 1450011. http://dx.doi.org/10.1142/s0219061314500111.

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Анотація:

We examine the sequences A that are low for dimension, i.e. those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of KA(n)/K(n) is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0.

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31

SVOZIL, KARL, and ANTON ZEILINGER. "DIMENSION OF SPACE-TIME." International Journal of Modern Physics A 01, no. 04 (December 1986): 971–90. http://dx.doi.org/10.1142/s0217751x86000368.

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Анотація:

In order to make it operationally accessible, it is proposed that the notion of the dimension of space-time be based on measure-theoretic concepts, thus admitting the possibility of noninteger dimensions. It is found then, that the Hausdorff covering procedure is operationally unrealizable because of the inherent finite space-time resolution of any real experiment. We therefore propose to define an operational dimension which, due to the quantum nature of the coverings, is smaller than the idealized Hausdorff dimension. As a consequence of the dimension of space-time less than four, relativistic quantum field theory becomes finite. Also, the radiative corrections of perturbation theory are sensitive on the actual value of the dimension 4–ε. Present experimental results and standard theoretical predictions for the electromagnetic moment of the electron seem to suggest a nonvanishing value for ε.

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32

Biś, Andrzej, and Agnieszka Namiecińska. "Hausdorff Dimension and Topological Entropies of a Solenoid." Entropy 22, no. 5 (April 28, 2020): 506. http://dx.doi.org/10.3390/e22050506.

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Анотація:

The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz solenoids and locally λ — expanding solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally λ — expanding solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of λ .

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33

Mendel, Manor. "A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces." Analysis and Geometry in Metric Spaces 10, no. 1 (January 1, 2022): 50–62. http://dx.doi.org/10.1515/agms-2022-0133.

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Abstract The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least β. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal’s Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of “nearly ultrametric” subsets of compact spaces having full Hausdorff dimension.

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34

ZHANG, YAN-FANG. "A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES." Fractals 28, no. 06 (September 2020): 2050115. http://dx.doi.org/10.1142/s0218348x20501157.

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Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.

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35

Essex, Christopher, and M. A. H. Nerenberg. "Fractal dimension: Limit capacity or Hausdorff dimension?" American Journal of Physics 58, no. 10 (October 1990): 986–88. http://dx.doi.org/10.1119/1.16262.

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36

LIU, JIA, and DEZHI LIU. "ON THE DECOMPOSITION OF CONTINUOUS FUNCTIONS AND DIMENSIONS." Fractals 28, no. 01 (December 31, 2019): 2050007. http://dx.doi.org/10.1142/s0218348x20500073.

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In this paper, we consider decomposition of continuous functions in [Formula: see text] in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers [Formula: see text], any real-valued continuous function in [Formula: see text] can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension [Formula: see text] and lower box dimension [Formula: see text]. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

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37

MARTÍNEZ, LUIS MANUEL, and GAMALIEL BLÉ. "HAUSDORFF DIMENSION OF JULIA SETS OF QUADRATIC POLYNOMIALS." Fractals 26, no. 03 (June 2018): 1850020. http://dx.doi.org/10.1142/s0218348x18500202.

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The Hausdorff dimension of Julia sets of expanding maps can be computed by the eigenvalue algorithm. In this work, an implementation of this algorithm for quadratic polynomial, that allows the calculation of the Hausdorff dimension of Julia sets for complex parameters, is done. In particular, the parameters in a neighborhood of the parabolic parameter [Formula: see text] are analyzed and a small oscillation in Hausdorff dimension is shown.

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38

Honda, Shouhei. "On low-dimensional Ricci limit spaces." Nagoya Mathematical Journal 209 (March 2013): 1–22. http://dx.doi.org/10.1017/s0027763000010667.

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AbstractWe call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

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39

Guan, Lifan, and Ronggang Shi. "Hausdorff dimension of divergent trajectories on homogeneous spaces." Compositio Mathematica 156, no. 2 (December 19, 2019): 340–59. http://dx.doi.org/10.1112/s0010437x19007711.

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For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].

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40

NGAI, SZE-MAN. "MULTIFRACTAL STRUCTURE OF NON-COMPACTLY SUPPORTED INFINITE MEASURES." Fractals 16, no. 03 (September 2008): 209–26. http://dx.doi.org/10.1142/s0218348x0800396x.

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Many important definitions in the theory of multifractal measures on ℝd, such as the Lq-spectrum, L∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported or infinite measures. We propose definitions that extend the original definitions to positive Borel measures on ℝd which are finite on bounded sets, and recover many important results that hold for compactly supported finite measures. In particular, we prove that if the Lq-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.

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41

Anjana, V., E. Harikumar, and A. K. Kapoor. "Noncommutative space–time and Hausdorff dimension." International Journal of Modern Physics A 32, no. 31 (November 8, 2017): 1750183. http://dx.doi.org/10.1142/s0217751x17501834.

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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 [Formula: see text] and the resolution [Formula: see text] 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.

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42

YAYAMA, YUKI. "APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS." Stochastics and Dynamics 11, no. 04 (November 21, 2011): 643–79. http://dx.doi.org/10.1142/s0219493711003486.

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Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

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43

Tukia, Pekka. "Hausdorff dimension and quasisymmetric mappings." MATHEMATICA SCANDINAVICA 65 (December 1, 1989): 152. http://dx.doi.org/10.7146/math.scand.a-12274.

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44

Cheung, Yitwah, and Nicolas Chevallier. "Hausdorff dimension of singular vectors." Duke Mathematical Journal 165, no. 12 (September 2016): 2273–329. http://dx.doi.org/10.1215/00127094-3477021.

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45

Rushing, T. B. "Hausdorff dimension of wild fractals." Transactions of the American Mathematical Society 334, no. 2 (February 1, 1992): 597–613. http://dx.doi.org/10.1090/s0002-9947-1992-1162104-8.

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46

Pollicott, Mark. "Hausdorff dimension and asymptotic cycles." Transactions of the American Mathematical Society 355, no. 8 (April 16, 2003): 3241–52. http://dx.doi.org/10.1090/s0002-9947-03-03308-7.

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47

Aravinda, C. S. "Bounded geodesics and Hausdorff dimension." Mathematical Proceedings of the Cambridge Philosophical Society 116, no. 3 (November 1994): 505–11. http://dx.doi.org/10.1017/s0305004100072777.

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Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

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48

De Reyna, J. Arias. "Hausdorff dimension of Banach spaces." Proceedings of the Edinburgh Mathematical Society 31, no. 2 (June 1988): 217–29. http://dx.doi.org/10.1017/s0013091500003345.

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We show that if X is a Banach space of infinite dimension and μh is a Hausdorff measure, where h is continuous, then there exists a measurable set K ⊂ X such that 0<μh(K)< + ∞. We also characterize the normed spaces in which the unit ball can be covered by a sequence of balls whose radii rn < 1 converge to zero as n → ∞.

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BISWAS, KINGSHOOK. "Hedgehogs of Hausdorff dimension one." Ergodic Theory and Dynamical Systems 28, no. 06 (October 15, 2008): 1713. http://dx.doi.org/10.1017/s0143385707000879.

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50

Dufloux, Laurent. "Hausdorff dimension of limit sets." Geometriae Dedicata 191, no. 1 (March 25, 2017): 1–35. http://dx.doi.org/10.1007/s10711-017-0240-2.

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