Generalised L\"uroth expansions and a family of Minkowski's Question-Mark functions

TL;DR

This paper introduces a family of functions generalizing Minkowski's Question-Mark function using $eta$-L"uroth expansions, showing they are singular homeomorphisms that map quadratic surds to rationals, extending key properties of the original function.

Contribution

It constructs a new family of singular homeomorphisms generalizing Minkowski's Question-Mark function via $eta$-L"uroth expansions, revealing their conjugacy to the Gauss Map.

Findings

All functions are singular homeomorphisms of the interval.

They map quadratic surds to rational numbers in rational cases.

The family generalizes key properties of Minkowski's Question-Mark function.

Abstract

The Minkowski's Question-Mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the twentieth century. Using different representations of real numbers by infinite sequences of integers, called $\alpha$-L\"uroth expansions, we obtain different instances of the standard shift map on infinite symbols, all of them topologically conjugated to the Gauss Map. In this note we prove that each of these conjugations share properties with the Minkowski's Question-Mark function: all of them are singular homeomorphisms of the interval, and in the "rational" cases, they map the set of quadratic surds into the set of rational numbers. In this sense, this family is a natural generalisation of the Minkowski's Question-Mark function.