NEW PROOF OF THE AHLFORS FIVE ISLANDS THEOREM 339 from

Generation of the Ahlfors Five Islands Theorem

The Role of the Ahlfors Five Islands Theorem in Complex Dynamics

The Ahlfors five islands theorem has become an important tool in complex dynamics. We discuss its role there, describing how it can be used to deal with a variety of problems. This includes questions concerning the Hausdorff dimension of Julia sets, the existence of singleton components of Julia sets, and the existence of repelling periodic points. We point out that for many applications a simp...

An Ahlfors Islands Theorem for Non-archimedean Meromorphic Functions

We present a p-adic and non-archimedean version of Ahlfors’ Five Islands Theorem for meromorphic functions, extending an earlier theorem of the author for holomorphic functions. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed.

A new proof for the Banach-Zarecki theorem: A light on integrability and continuity

To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

The Basic Theorem and its Consequences

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...