In this paper, we prove that every $C^1$ generic diffeomorphism $f$ on a compact smooth manifold $M$ is partially hyperbolic if it has the pseudo-orbit tracing property. Moreover we show that every $C^1$ generic vector filed $\varphi$ on $M$ with $dim...
In this paper, we prove that every $C^1$ generic diffeomorphism $f$ on a compact smooth manifold $M$ is partially hyperbolic if it has the pseudo-orbit tracing property. Moreover we show that every $C^1$ generic vector filed $\varphi$ on $M$ with $dimM = 3$ is singular hyperbolic if it has the pseudo-orbit tracing property. This is a partial answer of the conjecture proposed by Abdenur and D\'iaz in \cite{AD}.