Abstract:Aims and scope of the seriesThe series 'Atlantis Studies in Mathematics for Engineering and Science'(AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of pro… Show more
“…Usual statements of the Hillam-Thron Theorem, such as [10, Theorem 4.37], have essentially the same hypotheses as Theorem 4.1, but they have only this weaker conclusion that S n converges on D to a constant. There are variants on the Hillam-Thron Theorem (such as [11,Lemma 3.8]) in which the hypothesis s n (v) = u is weakened, and our observations on sets of divergence apply to most, if not all, of these alternative theorems.…”
Section: Proof Of Theorem 11: Part Imentioning
confidence: 92%
“…The Seidel-Stern Theorem ([4, Theorem 1.8] or [11,Theorem 3.13]) states that, with our hypotheses, T n converges at 0 to a point p. Since T n (∞) = T n−1 (0) we also have that T n (∞) → p as n → ∞. Since T n converges to p at two distinct points we see from Lemma 3.1 that T n converges generally to p. Next, observe that t −1 n (−∞, 0) ⊆ (−∞, 0) for each n so that, by Lemma 3.2, the Julia set of T n is contained in [−∞, 0].…”
Abstract. A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.
“…Usual statements of the Hillam-Thron Theorem, such as [10, Theorem 4.37], have essentially the same hypotheses as Theorem 4.1, but they have only this weaker conclusion that S n converges on D to a constant. There are variants on the Hillam-Thron Theorem (such as [11,Lemma 3.8]) in which the hypothesis s n (v) = u is weakened, and our observations on sets of divergence apply to most, if not all, of these alternative theorems.…”
Section: Proof Of Theorem 11: Part Imentioning
confidence: 92%
“…The Seidel-Stern Theorem ([4, Theorem 1.8] or [11,Theorem 3.13]) states that, with our hypotheses, T n converges at 0 to a point p. Since T n (∞) = T n−1 (0) we also have that T n (∞) → p as n → ∞. Since T n converges to p at two distinct points we see from Lemma 3.1 that T n converges generally to p. Next, observe that t −1 n (−∞, 0) ⊆ (−∞, 0) for each n so that, by Lemma 3.2, the Julia set of T n is contained in [−∞, 0].…”
Abstract. A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.
“…The value of K(a n | 1) -that is, the limit of the sequence T n (0) -is necessarily contained in H (and it cannot be 0 or ∞). In fact, it is known that every element in H \{0, ∞} is the value of some such continued fraction (see, for example, [15,Thm. 3.47]).…”
Section: 32])mentioning
confidence: 99%
“…The theorem was extended by some of these authors in a number of subsequent papers including [14,21], and the statement of the theorem from [21] is recast in the books by Jones and Thron [11,Thm. 4.42] and Lorentzen and Waadeland [15,Thm. 3.43].…”
Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Möbius transformations. This geometric approach allows us to relate the Stern-Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Möbius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions.
“…Such domains are indicated in the complex plane, that if elements a k , b k of a continued fraction belong to these domains then the continued fraction converges. At first convergence domains for continued fractions we can find in papers of Worpitzky (1865), Pringsheim (1899) and Van Vleck (1901) [8].…”
For a branched continued fraction of a special form we propose the limit value set for the Worpitzky-like theorem when the element set of the branched continued fraction is replaced by its boundary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.