# American Institute of Mathematical Sciences

December  2012, 32(12): 4133-4147. doi: 10.3934/dcds.2012.32.4133

## Inducing and unique ergodicity of double rotations

 1 Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom

Received  June 2011 Revised  October 2011 Published  August 2012

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.
Citation: Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133
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