The ergodic shadowing property and homoclinic classes

Abstract

In this paper, we show that if a diffeomorphism satisfies a local star condition and it has the ergodic shadowing property then it is hyperbolic. MSC: 37C29; 37C50

Periodic shadowing and standard shadowing property

“…Thus the diffeomorphism does not have sinks and sources. It is known that a Morse-Smale diffeomorphism has the shadowing property, but the diffeomorphism contains sinks and sources [8], so it does not have the ergodic shadowing property. Lee in [8], introduced the local star condition and using it he proved the hyperbolicity of H f (p), whenever H f (p) satisfies the local star condition and f has the ergodic shadowing property on H f (p), where H f (p) is the transversal homoclinic class of f associated to hyperbolic periodic point p of saddle type.…”

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.…”