Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{\sigma }$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,\sigma }$ is continuous at σ₀ as the function of the parameter $\sigma \in \overline{₀}$ if and only if $HD\left(J_{0,\sigma ₀}\right)\ge 4/3$. Since $HD\left(J_{0,\sigma }\right)>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD\left(J_{0,\sigma }\right)$ on an open and dense subset of...

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

Let $Q={\left(q_n\right)}_{n=1}^{\infty }$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f\times g:X\to Y\times I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{\delta }$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and $D_j$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod }_{j=1}^k{D}_j\times I^{p+1-i}\right)$ is (i+1)-to-1 is a dense $G_{\delta }$-subset of $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ for each 0 ≤ i ≤ p.

Let ${fₙ}_{n\ge 1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ\left(x\right)}_{n\ge 1}$ of integers, called the digit sequence of x, such that $x=li{m}_{n\to \infty }{f}_{a₁\left(x\right)}\circ \cdots \circ f_{aₙ\left(x\right)}\left(1\right)$. We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $x\in \Lambda :aₙ\left(x\right)\in B\left(\forall n\ge 1\right),li{m}_{n\to \infty }aₙ\left(x\right)=\infty$ for any infinite subset B ⊂ ℕ, a question posed by...

We study questions related to exceptional sets of pluri-Green potentials $V_{\mu }$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{\mu }$ are defined by $V_{\mu }\left(z\right)={\int }_{B}log(1/|{\varphi }_z\left(w\right)\left|\right)d\mu \left(w\right)$, where for a fixed z ∈ B, ${\varphi }_z$ denotes the holomorphic automorphism of B satisfying ${\varphi }_z\left(0\right)=z$, ${\varphi }_z\left(z\right)=0$ and $\left({\varphi }_z\circ {\varphi }_z\right)\left(w\right)=w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

A new class of functions called “$R_z$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_z$-supercontinuous functions properly includes the class of $R_{\mathrm{cl}}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\mathrm{cl}$-supercontinuous ($\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...

Given a topological space $X$, let $𝒰X$ and ${\eta }_X:X\to 𝒰X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f:X\to Y$, there is a continuous function $𝒰f:𝒰X\to 𝒰Y$, called the Salbany lift of $f$, satisfying $\left(𝒰f\right)\circ {\eta }_X={\eta }_Y\circ f$. If a continuous function $f:X\to Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F:𝒰X\to Y$ of $f$, not necessarily unique, such that $F\circ {\eta }_X=f$. In this paper, we give a condition on a space such that its Salbany map is open. In...

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

In the moduli space ${ℳ}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_{\mathrm{bif}}$ which is called the bifurcation current. This current gives rise to a measure ${\mu }_{\mathrm{bif}}:={\left(T_{\mathrm{bif}}\right)}^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\mathrm{supp}\left({\mu }_{\mathrm{bif}}\right)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $(2d-2)$ distinct neutral cycles is dense in a set of full Hausdorff dimension.

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by $U\cap M:U\in \cap M$. Suppose $X_M$ is homeomorphic to the irrationals; must $X=X_M$? We have partial results. We also answer a question of Gruenhage by showing that if $X_M$ is homeomorphic to the “Long Cantor Set”, then $X=X_M$.

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_i\in 0,1$. We prove that for any x ∈ (0,1), (x) contains a sequence ${{\beta }_k}_{k\ge 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\overline{\left(x\right)}$ are nowhere dense.

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...