A strange continued fraction associated with the Romik map

TL;DR

This paper investigates the Romik dynamical system, exploring its properties, invariant measures, and a novel 'strange' continued fraction expansion with unique digit sets.

Contribution

It extends the understanding of the Romik system by analyzing its properties, invariant measures, and introducing a new 'strange' continued fraction expansion.

Findings

The Romik system's basic properties and expansions are characterized.

An explicit invariant measure for the Romik system is derived and shown to be ergodic.

A new 'strange' continued fraction with digits {0, ±2} is introduced and analyzed.

Abstract

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the ``Romik system''), is ergodic and has a $\sigma$-finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the $\sigma$-finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every $x$ asymptotically half of the regular continued fraction (RCF) convergents of $x$ are among the Romik convergents. We also show that related to the Romik map a ``strange'' continued fraction can be given. ``Strange,'' as the set of possible partial quotients (i.e., digits) for any $x\in [0,1]$ in this expansion is $\{ 0, \pm 2\}$. Various properties of this ``Romik expansion'' are given.