Geodesic
On Certain Hyperbolic Sets
Matematičeskie zametki, Tome 87 (2010) no. 5, pp. 650-668

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Two invariant sets $F$ of certain diffeomorphisms $S$ that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an $F$, there is no locally maximal set containing $F$ are considered. It is proved that this property, although referring to the behavior of the orbits of $S$ near $F$, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of $S$ on $F$ with the hyperbolicity of $F$. (This means that if a hyperbolic set $F'$ for a diffeomorphism $S'$ is equivariantly homeomorphic to a Fathi–Crovisier or a Fisher set, then $F'$ has a neighborhood in which $S'$ has no locally maximal set containing $F'$.)
Keywords: non–locally premaximal hyperbolic set, non–locally premaximal hyperbolic set, hyperbolic set, locally maximal (premaximal) invariant set, metric space. 
@article{MZM_2010_87_5_a1,
     author = {D. V. Anosov},
     title = {On {Certain} {Hyperbolic} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {650--668},
     publisher = {mathdoc},
     volume = {87},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_5_a1/}
}
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D. V. Anosov. On Certain Hyperbolic Sets. Matematičeskie zametki, Tome 87 (2010) no. 5, pp. 650-668. http://geodesic.mathdoc.fr/item/MZM_2010_87_5_a1/