A class of Newton maps with Julia sets of Lebesgue measure zero

Let $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

Relative bifurcation sets and the local dimension of univoque bases

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

Escaping orbits are rare in the quasi-periodic Fermi–Ulam ping-pong

We consider the quasi-periodic Fermi–Ulam ping-pong model with no diophantine condition on the frequencies and show that typically the set of initial data which leads to escaping orbits has Lebesgue measure zero.

On the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector

It is shown that the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector can be of Lebesgue measure zero and of first Baire category. This result gives an answer to a question posed by G. Lazou.

On The K-Pseudo Symmetric and Ordinary Differentiation

In 1972, S. Valenti introduced the definition of k-pseudo symmetric derivative and has shown that the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is of Lebesgue measure zero. In 1993, L. Zajícek has shown that for a continuous function f, the set of all points, at which f is symmetrically differentiable but no differentiable, is σ-(1 - ε) symmetrically porous for every ε > 0. The question arises: can we transferred the Zajícek’s result to the case of the k-pseudo symmetric derivative?In this paper, we shall show that for each 0 < ε < 1 the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is σ-(1 - ε)-porous.

Stability of generalized quadratic functional equation on a set of measure zero

In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equationwhere E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.

On the Boundary of Self-Affine Sets

This paper is devoted to studying the boundary behavior of self-affine sets. We prove that the boundary of an integral self-affine set has Lebesgue measure zero. In addition, we consider the variety of the boundary of a self-affine set when some other contractive maps are added. We show that the complexity of the boundary of the new self-affine set may be the same, more complex, or simpler; any one of the three cases is possible.