C2-Stably Limit Shadowing Diffeomorphisms

Abstract

Let $f$ be a diffeomorphism on a $C^{\infty }$ closed surface. In this paper, we show that if $f$ has the $C^2$-stably limit shadowing property, then we have the following: (i) $f$ satisfies the Kupka-Smale condition; (ii) if $\mathrm{P(f)}$ is dense in the nonwandering set $\Omega$($f$) and if there is a dominated splitting on $P_s$($f$), then $f$ satisfies both Axiom $A$ and the strong transversality condition.