Almost every interval translation map of three intervals is finite type

Denis Volk

doi:10.3934/dcds.2014.34.2307

Article Contents

2014, Volume 34, Issue 5: 2307-2314. Doi: 10.3934/dcds.2014.34.2307

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Almost every interval translation map of three intervals is finite type

Denis Volk^1,

1.
Kungliga Tekniska Hogskolan (Royal Institute of Technology), Department of mathematics, SE-100 44, Stockholm

Received: November 2012

Revised: August 2013

Published: May 2014

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Abstract

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.

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Mathematics Subject Classification: Primary: 37C05, 37C20, 37C70, 37D20, 37D45.

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