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 and 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.

[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.

show all references

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.

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