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: May 31, 2011

Revised: September 30, 2011

Published: November 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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References

[1]	M. Barnsley, "Fractals Everywhere,'' Academic Press Inc., 1988.
[2]	G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc., 85 (1957), 219-227.doi: 10.2307/1992971.
[3]	M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergod. Th. Dyn. Sys., 15 (1995), 821-831.doi: 10.1017/S0143385700009652.
[4]	H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math., 137 (2003), 125-148.doi: 10.1007/BF02785958.
[5]	J. Buzzi and P. Hubert, Piecewise monotone maps without periodic points: Rigidity, measures and complexity, Ergodic Theory Dynam. Systems, 24 (2004), 383-405.doi: 10.1017/S0143385703000488.
[6]	M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196.doi: 10.1007/BF03007668.
[7]	H. B. Keynes and D. Newton, A "minimal'', non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), 101-105.doi: 10.1007/BF01214699.
[8]	R. Mañé, "Ergodic Theory and Differentiable Dynamics,'' Springer-Verlag, 1987.
[9]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.doi: 10.2307/1971341.
[10]	W. de Melo and S. van Strien, "One-Dimensional Dynamics,'' Springer-Verlag, 1996.
[11]	H. Suzuki, S. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Sys., 13 (2005), 515-532.doi: 10.3934/dcds.2005.13.515.
[12]	J. Schmeling and S. Troubetzkoy, Interval translation mappings, in "Dynamical Systems From Crystals to Chaos,'' J.-M. Gambaudo et al. eds., World Scientific, Singapore, 2000, 291-302.
[13]	W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242.doi: 10.2307/1971391.