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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.

MSC:
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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