Geodesic
Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 1, pp. 15-24

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We construct an open set of $ C^2$-diffeomorphisms which preserve the boundary of some manifold, and which have the following property: for each mapping, the basin of attraction of one component of the attractor is open and everywhere dense, but the basin of attraction of the second component is nowhere dense, though its measure is positive. 
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     author = {N. A. Solodovnikov},
     title = {Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {15--24},
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     year = {2014},
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     url = {http://geodesic.mathdoc.fr/item/MMO_2014_75_1_a1/}
}
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N. A. Solodovnikov. Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 1, pp. 15-24. http://geodesic.mathdoc.fr/item/MMO_2014_75_1_a1/