Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable

Capture zones of the family of functions lambda z^m exp(z)

We consider the family of entire transcendental maps given by $F_{\lambda,m}= \lambda z^m exp(z)$ where m>=2. All functions $F_{\lambda,m}$ have a superattracting fixed point at z=0, and a critical point at z=-m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behaviour, i.e., \lambda values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then non-locally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.Comment: 25 pages, 14 figures. Accepted for publication in the International Journal of bifurcation and Chao

Singular values and bounded Siegel disks

Let f be an entire transcendental function of finite order and Delta be a forward invariant bounded Siegel disk for f with rotation number in Herman's class (Formula presented.). We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow

Local fixed point indices of iterations of planar maps

Let f : U →R2 be a continuous map, where U is an open subset of R2. We consider a fixed point p of f which is neither a sink nor a source and such that p is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations ind(fn, p) n=1 is periodic,bounded by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem [Annals of Math., 146 (1997), 241-293] onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere