Simultaneous dense and non-dense orbits for toral diffeomorphisms
2016 ◽
Vol 37
(4)
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pp. 1308-1322
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Keyword(s):
Open Set
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We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.
1982 ◽
Vol 86
(3)
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pp. 498-498
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1987 ◽
Vol 36
(2)
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pp. 283-287
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Keyword(s):
1986 ◽
Vol 22
(3)
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pp. 267-282
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Keyword(s):
1968 ◽
Vol 64
(3)
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pp. 585-588
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Keyword(s):
1975 ◽
Vol 51
(6)
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pp. 411-414
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