scholarly journals Linear fractional transformations and nonlinear leaping convergents of some continued fractions

2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Christopher Havens ◽  
Stefano Barbero ◽  
Umberto Cerruti ◽  
Nadir Murru
Keyword(s):  
Continued Fractions ◽  
Linear Fractional Transformations

Convergence of Continued Fractions

1968 ◽  
Vol 20 ◽  
pp. 1037-1055 ◽  
Author(s):  
William B. Jones ◽  
W. J. Thron
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Linear Fractional Transformations ◽  
Generating Sequence ◽  
Image Position

Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form1.1and let1.2The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).

scholarly journals Outer compositions of hyperbolic/loxodromic linear fractional transfomations

1992 ◽  
Vol 15 (4) ◽  
pp. 819-822 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Continued Fractions ◽  
Analytic Theory ◽  
Complex Numbers ◽  
Linear Fractional Transformations ◽  
Classical Means ◽  
Repelling Fixed Point ◽  
Attracting Fixed Point

It is shown, using classical means, that the outer composition of hyperbolic or loxodromic linear fractional transformations{fn}, wherefn→f, converges toα, the attracting fixed point off, for all complex numbersz, with one possible exception,z0. I.e.,Fn(z):=fn∘fn−1∘…∘f1(z)→αWhenz0exists,Fn(z0)→β, the repelling fixed point off. Applications include the analytic theory of reverse continued fractions.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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