Realizations of groups of piecewise continuous transformations of the circle

Yves Cornulier

doi:10.3934/jmd.2020003

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2020, Volume 16: 59-80. Doi: 10.3934/jmd.2020003

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Realizations of groups of piecewise continuous transformations of the circle

Yves Cornulier

CNRS and Univ Lyon, University Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne, France

Received: April 2019

Revised: September 15, 2019

Published: February 2020

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Abstract

We study the near action of the group $ \mathrm{PC} $ of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of $ \mathrm{PC} $ is said to be realizable if it can be lifted to a group of permutations of the circle.

We prove that every finitely generated abelian subgroup of $ \mathrm{PC} $ is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips.

The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.

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Mathematics Subject Classification: Primary: 37E05; Secondary: 20B27, 20F65, 22F05, 37C85.

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Figure 1. Examples of graphs of elements of $ \mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S}) $ (parameterizing the circle as an interval). The first belongs to $ \mathrm{IET}^+ $; the second belongs to $ \mathrm{IET}^- $; the third belongs to $ \mathrm{IET}^\bowtie \backslash\mathrm{IET}^\pm $. The fourth is a more "typical" element of $ \mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S}) $. The value at breakpoints is not prescribed, as we consider group elements as defined up to finite indeterminacy

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Figure 2. Graphs of a 132-flip and a triple flip: in each case there are two hyper-clean lifts, choosing either the endpoints denoted as circles or dots

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Figure 3. Graphs of $ u $, $ v $, $ w $ and $ s $

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