Related papers: A note on the topology of escaping endpoints

Let f be a transcendental meromorphic function. Suppose that the finite part of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded.…

We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an…

Let $f\colon X\to X$ be a dominant meromorphic self-map of a compact complex variety $X$ in the Fujiki class $\mathcal{C}$. If the topological degree of $f$ is strictly larger than the other dynamical degrees of $f$, we show that the number…

We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated…

Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set.…

The asymptotic behaviour of the solutions of Poincar\'e's functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring…

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable…

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost…

Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture…

We study groups of homeomorphic bijections on spaces that are finite unions of compact connected linearly ordered subsets. We prove that all such groups when endowed with the topology of point-wise convergence are topological groups. }

In this note we study some properties of topological entropy for non-compact non-metrizable spaces. We prove that if a uniformly continuous self-map $f$ of a uniform space has topological shadowing property then the map $f$ has positive…

Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko-Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show…

We introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and…

We prove that every wandering exposed Julia component of a rational map is to a singleton, provided that each wandering Julia component containing critical points is non-recurrent. Moreover, we show that the Julia set contains only finitely…

We study homeomorphisms of compact metric spaces whose restriction to the nonwandering set has the pseudo-orbit tracing property. We prove that if there are positively expansive measures, then the topological entropy is positive. Some short…

Let $f$ be an entire function with the form $f(z)=P(e^z)/e^z$, where $P$ is a polynomial with degree at least $2$ and $P(0)\neq 0$. We prove that the area of the complement of the fast escaping set (hence the Fatou set) of $f$ in a…

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape…

We study the asymptotic expansion of smooth one-dimensional maps. We give an example of an interval map for which the optimal shrinking of components exponential rate is not attained for any neighborhood of a certain fixed point in the…

In the affine fragment of continuous logic, type spaces are compact convex sets. I study some model theoretic properties of extreme types. It is proved that every complete theory $T$ has an extremal model, i.e. a model which realizes only…

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same…