Inducing and unique ergodicity of double rotations

Previous Article
On essential coexistence of zero and nonzero Lyapunov exponents
DCDS Home
This Issue
Next Article
Noninvertible cocycles: Robustness of exponential dichotomies

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

Abstract
Related Papers

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 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.

[1]	Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[2]	Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[3]	Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[4]	Marcello Trovati, Peter Ashwin, Nigel Byott. Packings induced by piecewise isometries cannot contain the arbelos. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 791-806. doi: 10.3934/dcds.2008.22.791
[5]	Arek Goetz. Dynamics of a piecewise rotation. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593
[6]	Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[7]	David Ralston, Serge Troubetzkoy. Ergodicity of certain cocycles over certain interval exchanges. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2523-2529. doi: 10.3934/dcds.2013.33.2523
[8]	Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[9]	Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004
[10]	François Ledrappier, Omri Sarig. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 411-433. doi: 10.3934/dcds.2006.16.411
[11]	Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969
[12]	Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[13]	Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
[14]	Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1389-1409. doi: 10.3934/dcds.2017057
[15]	Li Chen, Yongyan Sun, Xiaowei Shao, Junli Chen, Dexin Zhang. Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022131
[16]	Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[17]	Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[18]	Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure and Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493
[19]	Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547
[20]	Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Henk Bruin Gregory Clack
on Google Scholar
Henk Bruin Gregory Clack

[Back to Top]