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

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, 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.
Keywords: renormalisation, double rotation, unique ergodicity., interval translation map, Piecewise isometries.
Mathematics Subject Classification: 37B10, 37C70, 37D25, 37E05, 37E10, 58F1.
Citation: Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133
References:
[1]

M. Barnsley, "Fractals Everywhere,'' Academic Press Inc., 1988.  Google Scholar

[2]

G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc., 85 (1957), 219-227. doi: 10.2307/1992971.  Google Scholar

[3]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergod. Th. Dyn. Sys., 15 (1995), 821-831. doi: 10.1017/S0143385700009652.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.  Google Scholar

[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.  Google Scholar

[8]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,'' Springer-Verlag, 1987.  Google Scholar

[9]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200. doi: 10.2307/1971341.  Google Scholar

[10]

W. de Melo and S. van Strien, "One-Dimensional Dynamics,'' Springer-Verlag, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Barnsley, "Fractals Everywhere,'' Academic Press Inc., 1988.  Google Scholar

[2]

G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc., 85 (1957), 219-227. doi: 10.2307/1992971.  Google Scholar

[3]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergod. Th. Dyn. Sys., 15 (1995), 821-831. doi: 10.1017/S0143385700009652.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.  Google Scholar

[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.  Google Scholar

[8]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,'' Springer-Verlag, 1987.  Google Scholar

[9]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200. doi: 10.2307/1971341.  Google Scholar

[10]

W. de Melo and S. van Strien, "One-Dimensional Dynamics,'' Springer-Verlag, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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