Related papers: Examples in Discrete Iteration of Arbitrary Interv…

This paper investigates the dynamical behaviour of holomorphic self-maps of the upper half-plane. More precisely, we focus on the hyperbolic and parabolic self-maps whose orbits approach the Denjoy--Wolff point with the slowest possible…

This paper deals with semigroups of holomorphic self-maps of the upper half-plane that exhibit an extremal (i.e. the slowest possible) rate of convergence to their Denjoy--Wolff point. The main novelty lies in the parabolic case of zero…

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have…

Let $(\phi_t)$ be a continuous semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $\tau\in\overline{\mathbb{D}}$. We study the rate of convergence of the forward orbits of $(\phi_t)$ to the Denjoy-Wolff…

Let $F=(\phi, \psi):\mathbb{D}^2\to\mathbb{D}^2$ denote a holomorphic self-map of the bidisk without interior fixed points. It is well-known that, unlike the case with self-maps of the disk, the sequence of iterates $$\{F^n:=F\circ F\circ…

We prove that when $f$ is a continuous selfmap acting on compact metric space $(X,d)$ which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy. Using this fact…

An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers…

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the…

Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {\sl semigroup-fication}, which allows to construct a continuous semigroup $(\phi_t)$ of holomorphic self-maps of the unit disc whose time one map…

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for…

We study the behaviour of discrete dynamical systems generated by a continuous map $f$ of a compact real interval into itself where at randomly chosen times a function different from $f$ - so called impulse function is applied. We show that…

If $\phi$ is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy-Wolff Theorem predicts the existence of a point $p$ with $|p|\leq 1$ such that the iterates $\phi_{n}$ converge to $p$ uniformly on compact subsets of…

We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we…

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4,…

A complete characterization of parabolic self-maps of finite shift is given in terms of their Herglotz's representation. This improves a previous result due to Contreras, D\'iaz-Madrigal, and Pommerenke. We also derive some consequences for…

We study parabolic semigroups of finite shift in the unit disk with regard to the rate of convergence of their orbits to the Denjoy--Wolff point. We examine this rate in terms of Euclidean distance, hyperbolic distance and harmonic measure.…

In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This…

We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of…

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non…

We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a careful analysis of the Denjoy-Wolff points. In…