Entropy of Continued Fractions (Gauss-Kuzmin Entropy)

2011

The entropy $h(T_\alpha)$ of $\alpha$-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set $\EE$. We will exploit the explicit description of the fractal structure of $\EE$ to investigate the self-similarities displayed by the graph of the function $\alpha \mapsto h(T_\alpha)$. Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

2011

The entropy h(Tα) of α-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set E. We will exploit the explicit description of the fractal structure of E to investigate the self-similarities displayed by the graph of the function α → h(Tα). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

Mathematics, 2021

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.

Journal of Mathematical Sciences, 2007

The article is devoted to finite continued fractions for numbers a/b when integer points (a, b) are taken from a dilative region. Properties similar to the Gauss-Kuz'min statistics are proved for these continued fractions.

Ergodic Theory and Dynamical Systems, 2011

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued fractions, thus proving a conjecture of Nakada and Natsui.

arXiv: Number Theory, 2020

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Loch's theorem from 1964. Thus, we aimed to compare the effectiveness of Chan's continued fractions, $\theta-$expansions, $N-$continued fractions and Renyi-type continued fractions. A central role in fulfilling our goal is the entropy of the measure preserving transformations which generate these expansions.

Entropy, 2022

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Atlantis Studies in Mathematics for Engineering and Science, 2008

Aims and scope of the series The 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 progress in the disciplines mentioned above. All books in this series are co-published with World Scientific.

We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)continued fractions.

Monatshefte f�r Mathematik, 2001

We give an in®nite-order-chain representation of the sequence of the incomplete quotients of the grotesque continued fraction expansion. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to prove a Gauss± Kuzmin-type theorem for this expansion. Finally, we derive a two-dimensional Gauss±Kuzmin theorem and also obtain an estimate of the convergence rate.