On the F-expanding of Homoclinic class

Abstract

We establish a closing property for thin trapped homoclinic classes. Taking advantage of this property, we proved that if the homoclinic class $H(p)$ admits a dominated splitting $T_{H(p)}M=E{\oplus }_{<}F$, where $E$ is thin trapped (see Definition \ref{Def:TP}) and all periodic points homoclinically related to $p$ are uniformly $F$-expanding at the period (see Definition \ref{Def:expanding}), then $F$ is expanded (see Definition \ref{Def:TP}).