For a sequence of complex parameters \omega=(c_{m}) we consider the non-autonomous composition of functions P_{c_m}(z) = z^{n}+c_{m} z^{k} with n>k\geq 2 . The definitions of Julia, \mathcal{J}_{\omega} , and Fatou, \mathcal{F}_{\omega} , sets are naturally generalized to this setting. In this paper, we study the connectedness and some dynamic properties of the Julia set provided that the sequence \omega is randomly chosen from some bounded Borel set W\subset \mathbb{C} .
We show that all Cantor sets in \mathbb{R}^{d} can be accompanied by another Cantor set in \mathbb{R}^{d} so that their product has a pinned tree distance set with nonempty interior. As a corollary, we construct Cantor sets of Hausdorff dimension d/2 in \mathbb{R}^{d} for even d that have a pinned tree distance set with nonempty interior. Our results generalize to the setting in which the Euclidean distance, |x-y| , is replaced by a general function, \phi(x,y) , satisfying a mild derivative condition.
This paper establishes that the isometry group of a self-similar set in R-3 is finite, unless the set is contained in a line.
For a sequence of complex parameters omega = (c(m)) we consider the non-autonomous composition of functions P-cm(z) = z(n) + c(mz)(k) with n > k >= 2. The definitions of Julia, I-omega, and Fatou, F-omega, sets are naturally generalized to this setting. In this paper, we study the connectedness and some dynamic properties of the Julia set provided that the sequence omega is randomly chosen from some bounded Borel set W subset of C.
The alpha-local nondeterminism notion (alpha-LND, for short) has been introduced by the authors [Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023), 388-425] to investigate the existence, joint continuity, and uniform H & ouml;lder continuity, i.e., H & ouml;lder continuity in the time variable t uniformly in the space variable x, for local times L(x,t) of general processes. In the present paper, we aim to use the alpha-LND property to improve this uniform Holder continuity of local times to a uniform Besov regularity, i.e., Besov regularity in the time variable t uniformly in the space variable x and in p (the index of the Besov space B-p,infinity(nu)(I;R)). The Besov regularity of local times, in the time variable t for fixed space variable x, has never been treated in the literature even for Gaussian or stable processes. We also extend the classical Adler's theorem, i.e., Theorem 8.7.1 [in: The Geometry of Random Fields (1981)] to the Besov spaces case. These results are then exploited to study the uniform (in p) Besov irregularity of the sample paths of the underlying processes. As applications, we get sharp uniform Besov irregularity results for a class of Gaussian processes and the solutions of systems of non-linear stochastic heat equations. The uniform Besov regularity of their corresponding local times is also obtained.
We investigate modified Sierpiński Carpet fractals, constructed by dividing a square into a square n \times n grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we produce “dust type” carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the fractal zeta functions, first introduced by Michel Lapidus, Goran Radunović, and Darko Žubrinić in their book (2017), from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpiński Carpet modification of dust type.
Let A be a 2× 2 integral matrix with an eigenvalue of modulus strictly less than 1. Let T be the natural endomorphism on the torus 𝕋^2=ℝ^2/ℤ^2, induced by A. Given τ>0, let R_τ ={ x∈𝕋^2 : T^nx∈ B(x,e^-nτ) infinitely many n∈ℕ }. We calculated the Hausdorff dimension of R_τ, and also prove that R_τ has a large intersection property.
We investigate the distribution of the largest digit for a wide class of infinite parabolic Iterated Function Systems (IFSs) of the unit interval. Due to the recurrence to parabolic (neutral) fixed points, the dimension analysis of these systems becomes more delicate than that of uniformly contracting IFSs. We show that the Hausdorff dimensions of level sets associated with the largest digits are constantly equal to the Hausdorff dimension of the limit set of the IFS. This result is an analogue of Wu and Xu's theorem [Math. Proc. Cambridge Philos. Soc. 146 (2009), 207-212] on the regular continued fraction. Examples of application of our result include the backward (aka minus, or negative) continued fractions, even-integer continued fractions, and go beyond. Our main tool is a dimension theory for non-uniformly expanding Bernoulli interval maps with infinitely many branches.
We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the relationship between distinct dimensions. In particular, we identify an equivalent condition when the box and Assouad dimension coincide, and show that under this condition, the Hausdorff dimension attains the same value.
We introduce the notion of (abelian) similarity scheme, as a constructive model for topological self-similar fractals, in the same way in which the notion of iterated function system furnishes a constructive notion of self-similar fractals in a metric environment. At the same time, our notion gives a constructive approach to the Kigami-Kameyama notion of topological fractals, since a similarity scheme produces a topological fractal a la Kigami-Kameyama, and many Kigami-Kameyama topological fractals may be constructed via similarity schemes. Our scheme consists of objects X-0 (sic)X-Phi (1) (sic) Y (sic) X0, where X-0, X(1)and Y are compact Hausdorff spaces, the map Phi is continuous injective and the map pi is continuous surjective. This scheme produces a sequence X-n, n is an element of N, of compact Hausdorff spaces, X-n} embedded in Xn+1, and a compact Hausdorff space X-infinity giving a sort of injective limit space, which turns out to be self-similar. We observe that the space Y parametrises the generalised similarity maps, and finiteness of Y is not required.
We compute the Hausdorff dimension of the closure of the generalized Brunner-Sidki-Vieira group acting on the $m$-adic tree for $m\ge 2$, providing the first examples of self-similar topologically finitely generated closed subgroups of transcendental Hausdorff dimension in the group of $m$-adic automorphisms.
We completely characterize the asymptotic behavior of the covering number for strongly connected inhomogeneous graph-directed self-similar sets satisfying the strong open set condition.
In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.
We give a self-similar iterated function system \{S_{i}=r_{i} x+b_{i}\}_{i=1}^{M} satisfying open set condition. Let E be the self-similar set and \mu the self-similar measure generated by a probability vector (p_{i})_{i=1}^{M} . Let f(x)=\mu((-\infty, x]) , and we write N for the collection of \alpha -Hölder non-differentiable points of f . Suppose that p_{i} >r^{\alpha}_{i} , i=1,\ldots, M . We provide the Hausdorff, packing and box dimension formulas for the non-differentiable set N of f under the open set condition.
This paper presents a novel functional framework by defining and analyzing the \psi -Hilfer fractional Orlicz space \mathcal{O}_{G}^{\alpha, \gamma, \psi}(\Lambda, \mathbb{R}) . This space extends traditional function spaces by integrating fractional calculus with the flexibility of Orlicz spaces, allowing for a broader class of function growth conditions. A key contribution of this work is the qualitative analysis of this newly introduced space. We investigate fundamental structural properties such as reflexivity, completeness, and separability, which play a crucial role in functional analysis and the study of variational problems. Additionally, we establish a continuous embedding of \mathcal{O}_{G}^{\alpha, \gamma, \psi}(\Lambda, \mathbb{R}) into a suitable Orlicz spaces. This result provides deeper insight into the relationship between our proposed space and existing functional frameworks, ensuring its applicability in mathematical analysis. As a practical application, we employ Ricceri’s three critical points theorem to demonstrate the existence of three weak solutions for a class of fractional Kirchhoff type equations. This application underscores the effectiveness of our newly developed space in solving variational problems, particularly in the context of critical point theory. Overall, this work bridges fractional calculus, Orlicz spaces, and variational analysis, providing a novel mathematical setting for studying complex differential equations.
The aim of this paper is to investigate fractals and their Hausdorff dimensions in a non-Archimedean setting. We are interested in fractals like equilibrated and fundamental sets of the Tate fields and, in particular, in fractals like Galois orbits of generic elements of infinite Galois extensions of p -adic fields. Also, we study the invariance of the Hausdorff dimension of Galois orbits defined by integral transcendental elements of \mathbb{C}_{p} .
For each integer k > 0, let n(k) and m(k) be integers such that n(k) >= 2, m(k) > 2, and let D-k be a subset of {0, ... , n(k)- 1} x {0, ... , m(k)- 1}. For each w = (i, j ) E D-k, we define an affine transformation on R-2 by Phi(w)(x) = T-k(x + w), w is an element of D-k, where T-k = diag(n(k)(-1) , m(k)(-1) ). The non-empty compact set E= boolean AND(infinity)(k=1)k=1 (w(1)w(2)...w(k))/is an element of Pi(i)(k=1) D-i is called a nonautonomous carpet. In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the nonautonomous carpets E. We also explore the dimension properties of nonautonomous measures mu supported on E, and we provide Hausdorff, packing and entropy dimension formulas of mu.
Given a compact set E \subset \mathbb{R}^{d} , we investigate for which values of m the equality \operatorname{dim}_{\theta} P_{V}(E) = m or \operatorname{dim}_{\theta} P_{V}(E) = \operatorname{dim}_{\theta} E holds for \gamma_{d,m} -almost all V \in G(d,m) . Our result extends to more general functions, including orthogonal projections and fractional Brownian motion. As a particular case, when \theta = 1 , the results apply to the box dimension.
