Existence of a Non-Uniquely Ergodic Interval Exchange Transformation with Flips Possessing Three Invariant Measures

TL;DR

This paper provides the first explicit example of an interval exchange transformation with flips that has three distinct invariant ergodic measures, advancing understanding of their ergodic properties.

Contribution

It introduces the first explicit example of a FIET with three invariant measures using a generalized Rauzy induction method.

Findings

First explicit example of a FIET with three invariant measures

Uses generalized Rauzy induction for proof

Contributes to ergodic theory of FIETs

Abstract

We present the first explicit example of an interval exchange transformation with flips (FIET) possessing three distinct invariant ergodic measures. The proof is based on a generalization of M. Keane's method, using the Rauzy induction adapted for FIETs, which contributes to the study of the ergodic properties of this class of dynamical systems.