Measure expansive homoclinic classes for generic diffeomorphisms

Abstract

Let f : M → M be a diffeomorphism on a closed smooth n(n ≥ 2)dimensional Riemannian manifold M. For C 1 generic f, if a homoclinic class H f (p) is measure expansive then it is hyperbolic.

Continuum-wise expansive homoclinic classes for robust dynamical systems

“…We can obtain the results for the R-robustly expansive homoclinic classes. According to these results, the following is a general result of [17].…”

“…Sambarino and Vieitez [26] proved that if the homoclinic class H(p, f ) is generically C 1 -robustly expansive, then it is hyperbolic. Lee [17] proved that if a locally maximal homoclinic class H(p, f ) is homogeneous, then it is hyperbolic. Lee [16] proved that if a homoclinic class H(p, f ) is continuum-wise expansive, then it is hyperbolic.…”

“…Lee [17] proved that if the homoclinic class H(p, f ) is R-robustly measure expansive, then it is hyperbolic. We can obtain the results for the R-robustly expansive homoclinic classes.…”

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.…”

R-robustly measure expansive homoclinic classes are hyperbolic

“…Lee and Lee [9] proved that if the homoclinic class H f (p) is C 1 stably measure expansive then it is hyperbolic. Koo et al [6] proved that for C 1 generic f, if a locally maximal homoclinic class H f (p) is measure expansive, then it is hyperbolic.…”