منابع مشابه
Conical Limit Sets and Continued Fractions
Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, S. By identifying S with the boundary of three-dimensional hyperbolic space, H, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H. Usin...
متن کاملContinued Fractions with Three Limit Points
The research described in this paper was motivated by an enigmatic entry in Ramanujan’s lost notebook [11, p. 45] in which he claimed, in an unorthodox fashion, that a certain q-continued fraction possesses three limit points. More precisely, he claimed that as n tends to ∞ in the three residue classes modulo 3, the nth partial quotients tend, respectively, to three distinct limits, which he ex...
متن کاملContinued fractions built from convex sets and convex functions
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the ...
متن کاملHausdorff dimension of sets of divergence arising from continued fractions
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 dete...
متن کاملPeriodic Continued Fractions And
We investigate when an algebraic function of the form φ(λ) = −B(λ)+ √ R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coefficients in C, can be written as a periodic α-fraction of the form
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ژورنال
عنوان ژورنال: Conformal Geometry and Dynamics of the American Mathematical Society
سال: 2007
ISSN: 1088-4173
DOI: 10.1090/s1088-4173-07-00169-5