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Symbolic extensions and partially hyperbolic diffeomorphisms. (English) Zbl 1220.37016
Authors’ abstract: “We show there are no symbolic extensions \(C^1\)-generically among diffeomorphisms containing nonhyperbolic robustly transitive sets with a center indecomposable bundle of dimension at least 2. Similarly, \(C^1\)-generically homoclinic classes with a center indecomposable bundle of dimension at least 3 that satisfy a technical assumption called index adaption have no symbolic extensions.”
The results of the authors in this paper are a step towards proving the (not yet proved) principle that Diffeomorphisms with a splitting \(E^s \oplus E^c \oplus E^u\) such that \(E^c\) is nonhyperbolic and splits into one dimensional subbundles have symbolic extensions, while they generically do not have symbolic extensions if \(E^c\) splits into undecomposable subbundles of dimension at least 2.

37D30 Partially hyperbolic systems and dominated splittings
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C20 Generic properties, structural stability of dynamical systems
37B10 Symbolic dynamics
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