Related papers: Geometric Limits of Julia Sets with Parameters on …
We use the methods developed with M. Lyubich for proving complex bounds for real quadratics to extend E. De Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for…
We consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions. Notably, well-known results in the…
Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map $z\mapsto z^d+\alpha$ is not…
The main interest of this article is the one-sided boundedness of the local discrepancy of $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ on the interval $(0,c)\subset(0,1)$ defined by \[D_n(\alpha,c)=\sum_{j=1}^n 1_{\{\{j\alpha\}<c\}}-cn.\] We…
If $f$ is a transcendental entire function with only algebraic singularities we calculate the Ruelle operator of $f$. Moreover, we prove both (i) if $f$ has a summable critical point, then $f$ is not structurally stable under certain…
A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there is a sequence of intervals $(J_n)_{n\in\omega}$ covering $M$ and such that $|J_n|\leq \varepsilon^{n+1}$ for each $n\in\omega$. We show that there is a microscopic…
Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context,…
We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This…
Metrizable spaces are studied in which every closed set is an $\alpha$-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of…
For a positive integer $n$, a graph with at least $n$ vertices is $n$-existentially closed or simply $n$-e.c. if for any set of vertices $S$ of size $n$ and any set $T\subseteq S$, there is a vertex $x\not\in S$ adjacent to each vertex of…
The Hausdorff dimensions of the Julia sets for non-analytic maps: f(z) = z^2 + epsilon z^* and f(z) = {z^*}^2 + epsilon are calculated perturbatively for small epsilon. It is shown that Ruelle's formula for Hausdorff dimensions of analytic…
This paper discusses a more general contractive condition for a class of extended cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same…
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…
We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials.…
We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit…
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Ch\'eritat. Since then many efforts have been devoted to finding out new…
In this article, we investigate the boundary of the escaping set I(f) for quasiregular mappings on R^n, both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that the boundary of I(f) is the Julia set…
We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier under some addition condition (the position of the Denjoy - Wolff point) it was shown in [M.D.Contreras, S.Diaz-Madrigal and…
We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a…