Invariant measures for continued fraction algorithms with finitely many digits

TL;DR

This paper studies continued fraction expansions with finitely many digits on various intervals, deriving invariant measures and analyzing entropy behavior through natural extension and approximation methods.

Contribution

It introduces methods to find invariant measures for CF expansions with finitely many digits on non-standard intervals and analyzes entropy variations.

Findings

Invariant measure densities are obtained for several CF systems.

Entropy as a function of parameter α is estimated for specific N-expansions.

Numerical results reveal interesting entropy behavior patterns.

Abstract

In this paper we consider continued fraction (CF) expansions on intervals different from $[0,1]$. For every $x$ in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss-Kuzmin-L\'evy based approximation method is used. Finally, a subfamily of the $N$-expansions is studied. In particular, the entropy as a function of a parameter $\alpha$ is estimated for $N=2$ and $N=36$. Interesting behavior can be observed from numerical results.