Related papers: Non-escaping endpoints do not explode
Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $\omega$. Such sets…
Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding.…
We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative H\'enon maps which…
The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher…
We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.
We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$,…
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…
We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set.
We develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the…
In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet
Several fundamental properties of real complex networks, such as the small-world effect, the scale-free degree distribution, and recently discovered topological fractal structure, have presented the possibility of a unique growth mechanism…
We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for…
Let $f$ be a polynomial-like mapping of the sphere of degree $d \geq 2$. We show that the Julia set $J(f)$ of $f$ cannot be the union of a finite number of proper indecomposable subcontinua. As a corollary, we prove that $J(f)$ is an…
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…
In this paper we study two classes of meromorphic functions previously studied by Mayer, Kotus, and Urba\'nski. In particular we estimate a lower bound for the Julia set and the set of escaping points for non-autonomous additive and affine…
For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a…
Let $X$ be a regular curve and let $f: X\to X$ be a monotone map. In this paper, nonwandering set of $f$ and the structure of special $\alpha$-limit sets for $f$ are investigated. We show that AP$(f)= \textrm{R}(f) =\Omega(f)$, where…
We consider the transcendental entire function $ f(z)=z+e^{-z} $, which has a doubly parabolic Baker domain $U$ of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infity, and for which the…
A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are…
We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalisation this is a one parameter…