We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases, this set is equivalent to the recurring set on the fractal.

We define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

Let $\unicode[STIX]{x1D703}$ be an irrational number and $\unicode[STIX]{x1D711}:\mathbb{N}\rightarrow \mathbb{R}^{+}$ be a monotone decreasing function tending to zero. Let

$$\begin{eqnarray}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <\unicode[STIX]{x1D711}(n),\text{for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$

$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D703}$

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

The purpose of this paper twofold. Firstly, we establish $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if $\unicode[STIX]{x1D707}$ is a self-similar measure satisfying the strong separation condition and $\unicode[STIX]{x1D707}$ is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of $\unicode[STIX]{x1D707}$ defined by

$$\begin{eqnarray}\bigg\{x\in \mathbb{R}^{d}\,\bigg|\,\lim _{r{\searrow}0}\frac{\log \unicode[STIX]{x1D707}(B(x,r))}{\log r}=\unicode[STIX]{x1D6FC}\bigg\}\end{eqnarray}$$

$\unicode[STIX]{x1D6F1}_{3}^{0}$

In the paper, we provide an effective criterion for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

Based on the abstract version of the Smital property, we introduce an operator $DS$ . We use it to characterise the class of semitopological abelian groups, for which addition is a quasicontinuous operation.

In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\rightarrow \unicode[STIX]{x1D6FD}x\;\text{mod}\;1$ . Let $\unicode[STIX]{x1D6F9}_{i}$ ( $i=1,2$ ) be two positive functions on $\mathbb{N}$ such that $\unicode[STIX]{x1D6F9}_{i}\rightarrow 0$ when $n\rightarrow \infty$ . We determine the Lebesgue measure and Hausdorff dimension for the $\limsup$ set

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}_{1},\unicode[STIX]{x1D6F9}_{2})=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-x_{0}|<\unicode[STIX]{x1D6F9}_{1}(n),|T_{\unicode[STIX]{x1D6FD}}^{n}y-y_{0}|<\unicode[STIX]{x1D6F9}_{2}(n)\text{ for infinitely many }n\in \mathbb{N}\}\end{eqnarray}$$

$x_{0},y_{0}\in [0,1]$

For every $q\in (0,1)$ , we obtain the Herglotz representation theorem and discuss the Bieberbach problem for the class of $q$ -convex functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$ . In addition, we consider the Fekete–Szegö problem and the Hankel determinant problem for the class of $q$ -starlike functions, leading to two conjectures for the class of $q$ -starlike functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$ .

For a prescribed set of lacunary data with equally spaced knot sequence in the unit interval, we show the existence of a family of fractal splines satisfying for v = 0, 1, … ,N and suitable boundary conditions. To this end, the unique quintic spline introduced by A. Meir and A. Sharma [SIAM J. Numer. Anal. 10(3) 1973, pp. 433-442] is generalized by using fractal functions with variable scaling parameters. The presence of scaling parameters that add extra “degrees of freedom”, self-referentiality of the interpolant, and “fractality” of the third derivative of the interpolant are additional features in the fractal version, which may be advantageous in applications. If the lacunary data is generated from a function Φ satisfying certain smoothness condition, then for suitable choices of scaling factors, the corresponding fractal spline satisfies , as the number of partition points increases.

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.

As an application of the boundary parametrization developed in our previous papers, we propose a new method to deduce information on the connected components of the interior of tiles. This gives a systematic way to study the topology of a certain class of self-affine tiles. An example due to Bandt and Gelbrich is examined to prove the efficiency of the method.

Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$ , it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$ , called the digit sequence of $x$ , in the sense that

$$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$

In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

We give a new necessary and sufficient condition for an iterated function system to satisfy the deterministic chaos game. As a consequence, we give a new example of an iterated function system which satisfies the deterministic chaos game.

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

Let μ λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ 1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ λ shows that μ λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ 1/(2k). This is surprising because some spectra for the measure μ 1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ 1/(2k).

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D 1(ℝN ) when N ⩾ 3. The space resembles the standard Sobolev space H 1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L 1(RN ). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp (ℝN ) when N ⩾ 2 and 2 ⩽ p ⩽ N.