Usual limit shadowable homoclinic classes of generic diffeomorphisms

CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS

“…Chain components are natural candidates to replace Smale's hyperbolic basic set in nonhyperbolic theory of dynamical systems. Many recent papers (see [1,2,5,[11][12][13][16][17][18][19][20][21][22][23][24][25][26][27]), most of which are only for diffeomorphisms, explore their hyperbolic-like properties such as partial hyperbolicity and dominated splitting. For instance, in [13], Lee et al showed that if f has the C 1 -stably shadowing property on the chain components, then it is hyperbolic.…”

Lyapunov stable homoclinic classes for smooth vector fields

“…Ahn et al [3] proved that for generic C , if a di eomorphism f has the shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Lee [4] proved that for generic C , if a di eomorphism f has the limit shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Note that local maximality is quite a restrictive condition.…”

“…Arbieto et al [5] proved that for generic C , if a bi-Lyapunov stable homoclinic class is homogeneous and has the shadowing property, then it is hyperbolic. See [3,4,[6][7][8][9][10][11][12][13][14][15] for related results.…”

Volume-Preserving Diffeomorphisms with Various Limit Shadowing

“…In [Pilyugin, 1999], the authors proved that if a closed invariant set is hyperbolic for a diffeomorphism then the diffeomorphism has the limit shadowing property (see [Pilyugin, 1999, Theorem 1.41]). Recently, Lee [2012] showed that C 1 -generically, if a diffeomorphism has the limit shadowing property on the locally maximal homoclinic class, then it is hyperbolic. Finally, the weak limit shadowing property was introduced and studied in [Lee, 2001].…”