On the F-expanding of Homoclinic class

TL;DR

This paper proves that under certain conditions involving a thin trapped homoclinic class and a dominated splitting, the expanding subbundle F is indeed expanded, advancing understanding of homoclinic class dynamics.

Contribution

The paper introduces a closing property for thin trapped homoclinic classes and demonstrates F-expansion under specific dominated splitting conditions.

Findings

Established a closing property for thin trapped homoclinic classes.

Proved F is expanded if the class admits a dominated splitting with certain properties.

Connected periodic point behavior to the expansion of the F bundle.

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}).