American Institute of Mathematical Sciences

Advanced
April  1999, 5(2): 291-300. doi: 10.3934/dcds.1999.5.291

Topologically transitive homeomorphisms of quotients of tori

Grant Cairns 1, , Barry Jessup 2, and  Marcel Nicolau 3,

1. 

School of Mathematics, La Trobe University, Melbourne, Australia 3083, Australia
2. 

Department of Mathematics, University of Ottawa, Ottawa, Canada K1N6N5, Canada
3. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Revised  January 1998 Published  January 1999

This paper considers the following question: for what finite subgroups $G\subset GL(n, \mathbb Z)$, does there exist an element $A\in GL(n, \mathbb Z)$ inducing a topologically transitive homeomorphism of $T^n$/$G$ We show that for $n = 2$ and 3, the only possibility is $G =\{\pm I\}$. Curiously, in higher dimension the structure is less restrictive. We give a variety of examples in dimension 4. Nevertheless, we show that in dimension $\geq 4$, there are relatively few irreducible examples.
Keywords: chaos., Topologically transitive.
Mathematics Subject Classification: 54H20, 57S2.
Citation: Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[1]

Viorel Nitica. Examples of topologically transitive skew-products. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 351-360. doi: 10.3934/dcds.2000.6.351
[2]

Jan Kwiatkowski, Artur Siemaszko. Discrete orbits in topologically transitive cylindrical transformations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 945-961. doi: 10.3934/dcds.2010.27.945
[3]

Sergiĭ Kolyada, Mykola Matviichuk. On extensions of transitive maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 767-777. doi: 10.3934/dcds.2011.30.767
[4]

John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743
[5]

Salvador Addas-Zanata, Fábio A. Tal. Homeomorphisms of the annulus with a transitive lift II. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 651-668. doi: 10.3934/dcds.2011.31.651
[6]

Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945
[7]

Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251
[8]

Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819
[9]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[10]

Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 867-888. doi: 10.3934/dcds.2018037
[11]

Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771
[12]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
[13]

Mykola Matviichuk, Damoon Robatian. Chain transitive induced interval maps on continua. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 741-755. doi: 10.3934/dcds.2015.35.741
[14]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353
[15]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
[16]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
[17]

Maciej J. Capiński. Covering relations and the existence of topologically normally hyperbolic invariant sets. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 705-725. doi: 10.3934/dcds.2009.23.705
[18]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335
[19]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
[20]

Ming-Chia Li, Ming-Jiea Lyu. Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 243-252. doi: 10.3934/dcds.2011.30.243

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences