Ergodicity for $p$-adic continued fraction algorithms

TL;DR

This paper proves that a broad family of $p$-adic multidimensional continued fraction algorithms, including several well-known variants, are ergodic with respect to the Haar measure, extending understanding of their statistical properties.

Contribution

It establishes ergodicity for a large family of $p$-adic continued fraction algorithms, including several specific algorithms, generalizing previous results in the field.

Findings

All considered $p$-adic algorithms are ergodic with Haar measure.

Includes Schneider's, Ruban's, and $p$-adic Jacobi-Perron algorithms.

Extends ergodic theory to a broad class of multidimensional $p$-adic continued fractions.

Abstract

Following Schweiger's generalization of multidimensional continued fraction algorithms, we consider a very large family of $p$-adic multidimensional continued fraction algorithms, which include Schneider's algorithm, Ruban's algorithms, and the $p$-adic Jacobi-Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.