Distribution of periodic points of certain Gauss shifts with infinite invariant measure

TL;DR

This paper studies the distribution of periodic points in specific Gauss shifts related to continued fractions, revealing their connection to quadratic irrationals and their equidistribution with respect to infinite invariant measures.

Contribution

It identifies the sets of quadratic irrationals corresponding to periodic points in Gauss shifts and proves their equidistribution with respect to infinite invariant measures.

Findings

Periodic points coincide with E-reduced and B-reduced quadratic irrationals.

These irrationals are shown to be equidistributed under the invariant measures.

The work links continued fraction dynamics with measure-theoretic distribution of special quadratic irrationals.

Abstract

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (R\'enyi). We show that they coincide exactly with two sets of quadratic irrationals that we call $E$-reduced, and respectively $B$-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.