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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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