April  2013, 33(4): 1293-1296. doi: 10.3934/dcds.2013.33.1293

A note on equivalent definitions of topological transitivity

1. 

Department of Mathematics and Statistics, La Trobe University, Plenty Road Bundoora 3086, Victoria, Australia, Australia

Received  October 2011 Revised  December 2011 Published  October 2012

We show that a well known lemma concerning conditions equivalent to topological transitivity is false when posed in a setting that is too general. We also explore some ways of remedying this problem.
Citation: John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293
References:
[1]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[2]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics \textbf{1513}, 1513 (1992).   Google Scholar

[3]

J. de Vries, "Elements of Topological Dynamics,", Kluwer Academic Publishers, (1993).   Google Scholar

[4]

S. Kolyada and L. Snoha, Some aspects of topological transitivity -A survey,, in: L. Reich et al. (eds.), 334 (1997), 3.   Google Scholar

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P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

show all references

References:
[1]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[2]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics \textbf{1513}, 1513 (1992).   Google Scholar

[3]

J. de Vries, "Elements of Topological Dynamics,", Kluwer Academic Publishers, (1993).   Google Scholar

[4]

S. Kolyada and L. Snoha, Some aspects of topological transitivity -A survey,, in: L. Reich et al. (eds.), 334 (1997), 3.   Google Scholar

[5]

P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

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