Related papers: The escaping set in transcendental dynamics

Building on the work of Cassels we prove the existence of infinite families of compact orbits of the diagonal group in the space of lattices which accumulate only on the divergent orbit of the standard lattice. As a consequence, we prove…

This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate.…

We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.

Building on recent work by Rippon and Stallard, we explore the intricate structure of the spider's web fast escaping sets associated with certain transcendental entire functions. Our results are expressed in terms of the components of the…

In [arXiv:1904.12333], we introduced the concept of escaping set in general setting for a topological space and extended the notion of limit set and escaping set for the general semigroup generated by continuous self maps. In this paper we…

In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of…

The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical…

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…

Let $E_{\la}(z)=\la {\rm exp}(z), \ \lambda\in \mathbb C$ be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed $\la$, the set of points in $\mathbb…

We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental…

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that…

Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in…

The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering…

Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive…

We investigate the scaling of the escape rate from piecewise-linear dynamical systems displaying intermittency due to the presence of an indifferent fixed-point. Strong intermittent behaviour in the dynamics can result in the system…

A major open question in transcendental dynamics asks if it is possible for points in a wandering domain to have bounded orbits, and more strongly, for a wandering domain to iterate only in a bounded domain. In this paper we give a partial…

In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future…

The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal…

We first establish any continuum without interiors can be a limit set of iterations of an entire function on an oscillating wandering domain, and hence arise as a component of Julia sets. Recently, Luka Boc Thaler showed that every bounded…

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…