Diagonal points having dense orbit

T. K. Subrahmonian Moothathu

Colloquium Mathematicae, Tome 120 (2010), p. 127-138 / Harvested from The Polish Digital Mathematics Library

Résumé

Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).

Publié le : 2010-01-01

Zbl 1200.54020

EUDML-ID : urn:eudml:doc:286234

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     title = {Diagonal points having dense orbit},
     journal = {Colloquium Mathematicae},
     volume = {120},
     year = {2010},
     pages = {127-138},
     zbl = {1200.54020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm120-1-9}
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T. K. Subrahmonian Moothathu. Diagonal points having dense orbit. Colloquium Mathematicae, Tome 120 (2010) pp. 127-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm120-1-9/