Alternating $N$-expansions

TL;DR

This paper introduces a new family of continued fraction maps with cyclically varying digits, proves their convergence, studies their dynamical properties, and explores invariant measures, supported by simulations.

Contribution

It develops a novel class of continued fraction algorithms with cyclic digit variation, establishing convergence, invariant measures, and conjecturing explicit densities.

Findings

Proved convergence of the new continued fraction algorithms.

Established existence of a unique absolutely continuous invariant measure.

Supported conjecture of invariant density with simulations.

Abstract

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration.