Inducing and unique ergodicity of double rotations

Henk Bruin; Gregory Clack

doi:10.3934/dcds.2012.32.4133

Article Contents

2012, Volume 32, Issue 12: 4133-4147. Doi: 10.3934/dcds.2012.32.4133

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Inducing and unique ergodicity of double rotations

Henk Bruin^1, and
Gregory Clack^1,

1.
Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH

Received: June 2011

Revised: October 2011

Published: December 2012

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Abstract

In this paper we investigate ``double rotations'', i.e., interval translation maps that when considered on the circle, have just two intervals of continuity. Using the induction procedure described by Suzuki et al., we show that Lebesgue a.e. double rotation is of finite type, i.e., it reduces to an interval exchange transformation. However, the set of infinite type double rotations is shown to have Hausdorff dimension strictly between $2$ and $3$, and carries a natural induction-invariant measure. It is also shown that non-unique ergodicity of infinite type double rotations, although occurring, is a-typical with respect to every induction-invariant probability measure in parameter space.

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Mathematics Subject Classification: 37B10, 37C70, 37D25, 37E05, 37E10, 58F11.

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