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

Fatou’s Associates

Suppose that f is a transcendental entire function, V⊊C is a simply connected domain, and U is a connected component of f-1(V). Using Riemann maps, we associate the map f : U→V to an inner function g : D→D. It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain

AVALIAÇÃO DO VOLUME CORRENTE DE AR EM CÃES SUBMETIDOS A TORACOTOMIA EM BLOCO

A freqüência respiratória e o volume corrente de ar foram medidos em cães submetidos a técnica de toracotomia em bloco. Os animais foram tranqüilizados com acetilpromazina e anestesiados com tiopental sódico, acompanhado de entubação endotraqueal e utilização de mini-respirador automático à pressão positiva intermitente. Os valores do volume-minuto, freqüência respiratória e volume corrente foram registrados nos seguintes tempos: antes da indução de anestesia (T0), no final da cirurgia (T1), 24 horas (T2) e 7 dias depois do término da cirurgia (T3), respectivamente. Nos animais do grupo II, no final da cirurgia (Tempo 1) e 24 horas após (Tempo 2), foi feita anestesia local infiltrativa dos nervos intercostais junto das costelas seccionadas para comparação das prováveis alterações da mecânica respiratória e do volume corrente de ar no período pós-operatório. A técnica da toracotomia em bloco não provocou alterações da mecânica respiratória durante o período pós-operatório

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

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy-Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences ( F n ) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting. Our methods apply in particular to non -autonomous dynam- ical systems, when ( F n ) are forward compositions of holo- morphic maps, and to the study of wandering domains in holomorphic dynamics. The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a 'weak independence' version of the second Borel-Cantelli lemma and the concept from ergodic theory of 'shrinking targets'. (c) 2024 Published by Elsevier Inc