July  2009, 3(3): 379-405. doi: 10.3934/jmd.2009.3.379

Discontinuity-growth of interval-exchange maps

1. 

Department of Mathematics and Statistics, University of Michigan - Dearborn, 4901 Evergreen Rd., Dearborn,MI 48128, United States

Received  November 2008 Revised  May 2009 Published  August 2009

For an interval-exchange map $f$, the number of discontinuities $d(f^n)$ either exhibits linear growth or is bounded independently of $n$. This dichotomy is used to prove that the group $\mathcal{E}$ of interval-exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval-exchanges. As a further application of this dichotomy, a classification of centralizers in $\mathcal{E}$ is given. This classification is used to show that $\text{Aut}(\mathcal{E}) \cong \mathcal{E}$ $\mathbb{Z}$/$ 2 \mathbb{Z}$.
Citation: Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[1]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015

[2]

Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483

[3]

Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123

[4]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

[5]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130

[6]

Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089

[7]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437

[8]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113

[9]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008

[10]

Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2001-2029. doi: 10.3934/dcds.2020350

[11]

Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099

[12]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[13]

Mohamed Baouch, Juan Antonio López-Ramos, Blas Torrecillas, Reto Schnyder. An active attack on a distributed Group Key Exchange system. Advances in Mathematics of Communications, 2017, 11 (4) : 715-717. doi: 10.3934/amc.2017052

[14]

Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139

[15]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699

[16]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[17]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[18]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195

[19]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320

[20]

Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365

2021 Impact Factor: 0.641

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]