C¹-stably shadowable chain components

Abstract

Let p be a hyperbolic periodic saddle of a diffeomorphism f on a closed C ∞

R-robustly measure expansive homoclinic classes are hyperbolic

“…Then by Franks lemma, there is g ∈ G such that for any small γ > 0 we can construct a closed small curve I q containing q or a closed small circle C q centered at q such that I q ⊂ C g (p g ) and two endpoints are related to p g and C q ⊂ C g (p g ). Note that I q and C q are g π(q) -invariant, normally hyperbolic, and g lπ(q) | I q is the identity map for some l > 0 (see [18]). For I q , we define a measure µ ∈ M * (M) by…”

Shadowable Chain Components and Hyperbolicity

“…Many recent papers have explored their hyperbolic-like properties such as dominated splitting, partial hyperbolicity, etc. For instance, Sakai [9] proved that if the chain component C f (p) of a diffeomorphism f containing a hyperbolic periodic point p is C 1 -robustly shadowable and the C f (p)-germ of f is expansive, then C f (p) is hyperbolic. Moreover Wen et al [11] claimed that the assumption of the C f (p)-germ expansivity of f can be dropped in the above result to show the hyperbolicity of the C 1 -robustly shadowable chain component C f (p).…”

Hyperbolicity of Chain Transitive Sets With Limit Shadowing

“…In this paper we show that any chain transitive set of a diffeomorphism on a compact C ∞ -manifold which is C 1 -stably limit shadowable is hyperbolic. Moreover, it is proved that a locally maximal chain transitive set of a C 1 -generic diffeomorphism is hyperbolic if and only if it is limit shadowable.Transitive sets, homoclinic classes and chain components of a diffeomorphism are natural candidates to replace the hyperbolic basic sets in nonhyperbolic theory of differentiable dynamical systems, and many recent papers explored their "hyperbolic-like" properties (for more details, see [2,6,8,9,14,15]). …”

“…Transitive sets, homoclinic classes and chain components of a diffeomorphism are natural candidates to replace the hyperbolic basic sets in nonhyperbolic theory of differentiable dynamical systems, and many recent papers explored their "hyperbolic-like" properties (for more details, see [2,6,8,9,14,15]). …”