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In this paper, we estimate the oscillation of harmonic functions on the (N − 1)-dimensional Sierpiński gasket KN, where K3 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 KN are log2N. [ABSTRACT FROM AUTHOR]

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time τ and the fractal dimension of time-scale β. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations. [ABSTRACT FROM AUTHOR]

Denote by S n (x , y) the length of the longest common substring of x and y with shifts in their first n digits of the b-ary expansions. We show that the sets of pairs (x, y), for which the growth rate of S n (x , y) is α log n with 0 ≤ α ≤ ∞ , have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices. [ABSTRACT FROM AUTHOR]

In statistical applications, it is common to encounter parameters supported on a varying or unknown dimensional space. Examples include the fused lasso regression, the matrix recovery under an unknown low rank, etc. Despite the ease of obtaining a point estimate via optimization, it is much more challenging to quantify their uncertainty. In the Bayesian framework, a major difficulty is that if assigning the prior associated with a p-dimensional measure, then there is zero posterior probability on any lower-dimensional subset with dimension d < p. To avoid this caveat, one needs to choose another dimension-selection prior on d, which often involves a highly combinatorial problem. To significantly reduce the modeling burden, we propose a new generative process for the prior: starting from a continuous random variable such as multivariate Gaussian, we transform it into a varying-dimensional space using the proximal mapping. This leads to a large class of new Bayesian models that can directly exploit the popular frequentist regularizations and their algorithms, such as the nuclear norm penalty and the alternating direction method of multipliers, while providing a principled and probabilistic uncertainty estimation. We show that this framework is well justified in the geometric measure theory, and enjoys a convenient posterior computation via the standard Hamiltonian Monte Carlo. We demonstrate its use in the analysis of the dynamic flow network data. for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of special homogeneous Moran sets which is called the $ \left \{m_{k}\right \} $ { m k } -homogeneous Moran sets by the connected components, and obtain the Hausdorff dimension and the upper box dimension of the $ \left \{m_{k}\right \} $ { m k } -homogeneous Moran sets under some conditions. [ABSTRACT FROM AUTHOR]

We introduce the exact approximation order in the dynamics of $ \beta $-expansions which has its analogy in classic Diophantine approximation. More precisely, let $ E_{\beta}(\psi) $ be the set of real numbers in [0, 1) which are approximable by their convergents in $ \beta $-expansions to order $ \psi $ but to no better order. The Hausdorff dimension of $ E_{\beta}(\psi) $ is given for any monotonic function $ \psi $. [ABSTRACT FROM AUTHOR]

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field. [ABSTRACT FROM AUTHOR]

Let α , β ∈ [ 0 , 1) = R / Z such that at least one of them is irrational. It is known that the random αβ -orbits of the Bernoulli mixture of rotations of α and β by choosing them with equal probability 1 2 are uniformly distributed modulo 1 with probability one. Chen et al. (2021) [2] showed that the exceptional set in the probability space has full Hausdorff dimension. In this note, we prove that certain sets related to random αβ -orbits which are not dense in R / Z have Hausdorff dimension 1. Our result can be viewed as an improvement of the dimension result obtained by Chen, Wang and Wen. [ABSTRACT FROM AUTHOR]

Let f : M → M be a continuous map on a compact metric space M equipped with a fixed metric d, and let τ be the topology on M induced by d. We denote by M (τ) the set consisting of all metrics on M that are equivalent to d. Let mdim M (M , d , f) and mdim H (M , d , f) be, respectively, the metric mean dimension and mean Hausdorff dimension of f. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that mdim M (M , d , f) and mdim H (M , d , f) depend on the metric d chosen for M . In this work, we will prove that, for a fixed dynamical system f : M → M , the functions mdim M (M , f) : M (τ) → R ∪ { ∞ } and mdim H (M , f) : M (τ) → R ∪ { ∞ } are not continuous, where mdim M (M , f) (ρ) = mdim M (M , ρ , f) and mdim H (M , f) (ρ) = mdim H (M , ρ , f) for any ρ ∈ M (τ) . Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function. [ABSTRACT FROM AUTHOR]

We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154(1) (2001), 45-96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira's ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z. 303 (2023), 3], to prove that under the right hypothesis for the Cantor sets K1, . . ., Kn and the function h: Cn → Rl, the following formula holds: HD(h(K1 × K2 × · · ·×Kn)) = min{l, HD(K1) + · · ·+HD(Kn)}. [ABSTRACT FROM AUTHOR]

We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity L N (ω) which describes the maximal length of string of zeros of the prefix among the first N iterations of ω in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

Let 0 < \beta \leq \alpha < 1 and \kappa > 0. I prove that there exists \eta > 0 such that the following holds for every pair of Borel sets A,B \subset \mathbb {R} with \dim _{\mathrm {H}} A = \alpha and \dim _{\mathrm {H}} B = \beta: \begin{equation*} \dim _{\mathrm {H}} \{c \in \mathbb {R}: \dim _{\mathrm {H}} (A + cB) \leq \alpha + \eta \} \leq \tfrac {\alpha - \beta }{1 - \beta } + \kappa. \end{equation*} This extends a result of Bourgain from 2010, which contained the case \alpha = \beta. The paper also contains a \delta-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a_{1}B + \ldots + a_{n}B. [ABSTRACT FROM AUTHOR]

We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two. [ABSTRACT FROM AUTHOR]

For two real bases q 0 , q 1 > 1 , we consider expansions of real numbers of the form ∑ k = 1 ∞ i k / (q i 1 q i 2 ... q i k) with i k ∈ { 0 , 1 } , which we call (q 0 , q 1) -expansions. A sequence (i k) is called a unique (q 0 , q 1) -expansion if all other sequences have different values as (q 0 , q 1) -expansions, and the set of unique (q 0 , q 1) -expansions is denoted by U q 0 , q 1 . In the special case q 0 = q 1 = q , the set U q , q is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases (q 0 , q 1) with trivial U q 0 , q 1 from those with non-trivial U q 0 , q 1 is the graph of a function G (q 0) that we call generalized golden ratio. Similarly, the curve separating pairs (q 0 , q 1) with countable U q 0 , q 1 from those with uncountable U q 0 , q 1 is the graph of a function K (q 0) that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in q 0 and q 1 , that G and K are continuous, strictly decreasing, hence almost everywhere differentiable on (1 , ∞) , and that the Hausdorff dimension of the set of q 0 satisfying G (q 0) = K (q 0) is zero. We give formulas for G (q 0) and K (q 0) for all q 0 > 1 , using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems. [ABSTRACT FROM AUTHOR]

The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3, MnO2 and SrTiO3/MnO2 were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO2, PMMA-SrTiO3 and PMMA-SrTiO3/MnO2 composites. It was determined that PMMA-SrTiO3/MnO2 had the highest total absorbed energy value (Etot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO3/MnO2 agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO2 agglomerates, while the PMMA-MnO2 showed the lowest Etot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R2 value of 0.99486, showing the potential use of this approach in the optimization of composite materials' microstructure–property relationship. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the Hausdorff dimension of the fractal set x ∈ d : ∏ 1 ≤ i ≤ d | T β i n (x i) − x i | < ψ (n)   for infinitely many  n ∈ ℕ , where T β i is the standard β i -transformation with β i > 1 , ψ is a positive function on ℕ and | ⋅ | is the usual metric on the torus . Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let T be a d × d non-singular matrix with real coefficients. Then, T determines a self-map of the d -dimensional torus d : = ℝ d / ℤ d . For any 1 ≤ i ≤ d , let ψ i be a positive function on ℕ and Ψ (n) : = (ψ 1 (n) , ... , ψ d (n)) with n ∈ ℕ. We obtain the Hausdorff dimension of the fractal set { x ∈ d : T n (x) ∈ L (f n (x) , Ψ (n))   for infinitely many  n ∈ ℕ } , where L (f n (x , Ψ (n))) is a hyperrectangle and { f n } n ≥ 1 is a sequence of Lipschitz vector-valued functions on d with a uniform Lipschitz constant. [ABSTRACT FROM AUTHOR]

Let [ d 1 (x) , d 2 (x) , ... , d n (x) , ... ] be the Lüroth expansion of x ∈ (0 , 1 ] , and let L n (x) = max { d 1 (x) , ... , d n (x) }. It is shown that for any α ≥ 0 , the level set x ∈ (0 , 1 ] : lim n → ∞ L n (x) log log n n = α has Hausdorff dimension one. Certain sets of points for which the sequence { L n (x) } n ≥ 1 grows more rapidly are also investigated. [ABSTRACT FROM AUTHOR]

The problem of testing random number generators is considered and a new method for comparing the power of different statistical tests is proposed. It is based on the definitions of random sequence developed in the framework of algorithmic information theory and allows comparing the power of different tests in some cases when the available methods of mathematical statistics do not distinguish between tests. In particular, it is shown that tests based on data compression methods using dictionaries should be included in test batteries. [ABSTRACT FROM AUTHOR]

For integers a and $b\geq 2$ , let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$. The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q and $2 is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity. [ABSTRACT FROM AUTHOR]

The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that \begin{equation*} \dim _{\mathcal H}W(\psi)\times \cdots \times W(\psi)=d-1+\dim _{\mathcal H}W(\psi) \end{equation*} where W(\psi) is the set of \psi-well approximable points in \mathbb {R} and \psi : \mathbb {N}\to \mathbb {R}^+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand's famous inequality on the dimension of the Cartesian product of general sets. [ABSTRACT FROM AUTHOR]

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" of a "network" corresponding to a fractal set. This leads to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic 1-Hölder- α functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers (Buczolich et al., 2022 [9,10]). In this paper, we extend our investigations to the lower and upper box-counting dimensions as well: while the former yields results highly resembling the ones about the Hausdorff dimension of level sets, the latter exhibits a different behavior. Instead of "finding narrow-cross sections", results related to upper box-counting dimension "measure" how much level sets can spread out on the fractal, and how widely the generic function can "oscillate" on it. Key differences are illustrated by giving estimates concerning the Sierpiński triangle. [ABSTRACT FROM AUTHOR]

The purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys. 32 (5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$. In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$. Precisely, we extend the result of Fan et al , [Multifractal analysis of some multiple ergodic averages. Adv. Math. 295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$. [ABSTRACT FROM AUTHOR]

In this article, we manifest the existence of a new class of α -fractal functions without endpoint conditions in the space of continuous functions. Furthermore, we add the existence of the same class in numerous spaces such as the Hölder space, the convex Lipschitz space, and the oscillation space. We also estimate the fractal dimensions of the graphs of the newly constructed α -fractal functions adopting some function spaces and covering methods. [ABSTRACT FROM AUTHOR]

We consider a bounded strictly pseudoconvex domain Ω ⊂ C n with C 2 boundary. Then, we show that any compact Ahlfors–David regular subset of ∂ Ω of Hausdorff dimension β ∈ (0 , 2 n - 1 ] contains a peak set E of Hausdorff dimension equal to β . [ABSTRACT FROM AUTHOR]

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let be an -dimensional compact Riemannian manifold, where , and . In this paper, we construct a homeomorphism with mean Hausdorff dimension equal to . Furthermore, we prove that the set of homeomorphisms on with both lower and upper mean Hausdorff dimensions equal to is dense in . Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to contains a residual subset of [ABSTRACT FROM AUTHOR]

We extend several notions and results from the classical Patterson–Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

Let $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$. Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function $$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$ which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function. [ABSTRACT FROM AUTHOR]

We prove that, for every 0 ⩽ s ⩽ 1 , there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension–a computability-theoretic construct–and the point-to-set principle of J. Lutz and N. Lutz (2018). [ABSTRACT FROM AUTHOR]

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures. [ABSTRACT FROM AUTHOR]

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the "subset of positive and finite measure" property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the bivariate fractal interpolation function using the Rakotch contraction theory and iterated function system for a countable data set. We also give the existence of the invariant Borel probability measure supported on the graph of the bivariate fractal interpolation function. In particular, we highlight that our theory encompasses the bivariate fractal interpolation theory in both finite and countably infinite settings available in literature. [ABSTRACT FROM AUTHOR]

This paper intends to study the analytical properties of the Riemann–Liouville fractional integral and fractal dimensions of its graph on ℝ n . We show that the Riemann–Liouville fractional integral preserves some analytical properties such as boundedness, continuity and bounded variation in the Arzelá sense. We also deduce the upper bound of the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of Hölder continuous functions. Furthermore, we prove that the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of a function, which is continuous and of bounded variation in Arzelá sense, are n. [ABSTRACT FROM AUTHOR]

We introduce two notions of fractal sumset properties. A compact set K ⊂ R d is said to have the Hausdorff sumset property (HSP) if for any ℓ ∈ N ≥ 2 there exist compact sets K 1 , K 2 ,..., K ℓ such that K 1 + K 2 + ⋯ + K ℓ ⊂ K and dim H K i = dim H K for all 1 ≤ i ≤ ℓ . Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set K ⊂ R d is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in R d . [ABSTRACT FROM AUTHOR]

Suppose $ X $ is a compact connected metric space and $ f\colon X \to X $ is a metric coarse expanding conformal map in the sense of Haïssinsky-Pilgrim. We show that if $ X $ contains a homeomorphic copy of the letter 'Y', then the Hausdorff dimension of $ X $ is greater than one. As an application, we show that for a semi-hyperbolic rational map $ f $ its Julia set $ \mathcal{J}_f $ is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if $ \mathcal{J}_f $ is homeomorphic to a circle or a closed interval. [ABSTRACT FROM AUTHOR]

By iterating the Bolyai-Rényi transformation T(x) = (x + 1)2 (mod 1), almost every real number x ∈ [0, 1) can be expanded as a continued radical expression x = − 1 + x 1 + x 2 + ... + x n + ... with digits xn ∈ {0, 1, 2} for all n ∈ ℕ. For any real number n ∈ [0, 1) and digit i ∈ {0, 1, 2}, let rn(x, i) be the maximal length of consecutive i's in the first n digits of the Bolyai-Rényi expansion of x. We study the asymptotic behavior of the run-length function rn(x, i). We prove that for any digit i ∈ {0, 1, 2}, the Lebesgue measure of the set D (i) = { x ∈ [ 0 , 1) : lim n → ∞ r n (x , i) log n = 1 log θ i } is 1, where θ i = 1 + 4 i + 1 . We also obtain that the level set E α (i) = { x ∈ [ 0 , 1) : lim n → ∞ r n (x , i) log n = α } is of full Hausdorff dimension for any 0 ⩽ α ⩽ ∞. [ABSTRACT FROM AUTHOR]

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and lower bounds on the Hausdorff dimension of the residual set of a generalized Apollonian circle packing. This dimension (which also coincides with a critical exponent) is strictly greater than that of the Apollonian packing. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained. [ABSTRACT FROM AUTHOR]

For any real numbers x ∈ [ 0 , 1 ] and β > 1 , let r n (x , β) be the maximal length of consecutive 0's in the first n digits of the beta-expansion of x in base β. The run-length function r n (x , β) has been well studied for a fixed base β > 1 or a fixed real number x = 1. In this paper, we prove that for any x ∈ (0 , 1) , the set D x = { β > 1 : lim n → ∞ ⁡ r n (x , β) log β ⁡ n = 1 } is of full Lebesgue measure in (1 , + ∞). When the exceptional set is considered, we prove that for any real numbers 0 ≤ a ≤ b ≤ + ∞ , the set D x (a , b) = { β > 1 : lim inf n → ∞ r n (x , β) log β ⁡ n = a , lim sup n → ∞ r n (x , β) log β ⁡ n = b } is of full Hausdorff dimension. We also determine the Hausdorff dimension of the set F x (c , d) = { β > 1 : lim inf n → ∞ r n (x , β) n = c , lim sup n → ∞ r n (x , β) n = d } for any real numbers 0 ≤ c ≤ d ≤ 1. [ABSTRACT FROM AUTHOR]

We prove an estimate on the Hausdorff dimension of the set of two-sided boundary points of general Sobolev extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type. [ABSTRACT FROM AUTHOR]

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let be non-empty Borel sets. If X is not contained in any line, we prove that If dimHY>1, we have the following improved lower bound: Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}. While the results above concern , we also derive some counterparts in by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren. [ABSTRACT FROM AUTHOR]

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any $ s\in (0,+\infty ] $ s ∈ (0 , + ∞ ] , there is a function taken from this family with the order of growth s. Baker proved that the Hausdorff dimension of a transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth. [ABSTRACT FROM AUTHOR]

Let A = { A i } i = 1 ∞ be a sequence of sets with each A i being a non-empty collection of 0-1 sequences of length i. For x ∈ [ 0 , 1) , the maximal run-length function ℓ n (x , A) (with respect to A ) is defined to be the largest k such that in the first n digits of the dyadic expansion of x there is a consecutive subsequence in A k . Suppose that lim n → ∞ (log 2 | A n |) / n = τ for some τ ∈ [ 0 , 1 ] and one additional assumption holds, we prove a generalization of the Erdős–Rényi limit theorem which states that lim n → ∞ ℓ n (x , A) log 2 n = 1 1 - τ for Lebesgue almost all x ∈ [ 0 , 1) . For the exceptional sets, we prove under a certain stronger assumption on A that the set x ∈ [ 0 , 1) : lim n → ∞ ℓ n (x , A) log 2 n = 0 and lim n → ∞ ℓ n (x , A) = ∞ has Hausdorff dimension at least 1 - τ . [ABSTRACT FROM AUTHOR]

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time τ and the fractal dimension of time-scale β. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations.

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $ , we investigate the size of the uniform approximation set $$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y| The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $ -almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $ , where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $ , we show that for $ \mu _\phi $ -a.e. $ x $ , the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $. [ABSTRACT FROM AUTHOR]

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset X⊂Rm is a sequence Φ=({ϕi(j)}i∈I(j))j=1∞ of collections of uniformly contracting maps ϕi(j):X→X, where I(j) is a finite set. In comparison to usual iterated function systems, we allow the contractions ϕi(j) applied at each step j to depend on j. In this paper, we focus on a family of parameterized NIFSs on Rm. Here, we do not assume the open set condition. We show that if a d-parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of m and the Bowen dimension. Moreover, we give an example of a family {Φt}t∈U of parameterized NIFSs such that {Φt}t∈U satisfies the transversality condition but Φt does not satisfy the open set condition for any t∈U. [ABSTRACT FROM AUTHOR]

The set of all irreducible denominators of positive rationals whose continued fraction expansions consist of elements of the set is considered. We prove that, for any prime , the set contains almost all possible remainders on division by and the remainder term in the corresponding asymptotic formula decays according to a power law. [ABSTRACT FROM AUTHOR]