Continuum-wise expansive homoclinic classes for robust dynamical systems

Abstract

Abstract In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of C1 $C^{1}$-robustly expansive dynamical system. First, we show that if the homoclinic class H(p,f) $H(p, f)$, which contains a hyperbolic periodic point p, is R-robustly continuum-wise expansive, then it is hyperbolic. For a vector field, if the homoclinic class H(γ,X) $H(\gamma , X)$ does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic.

Keywords

• Expansive
• Measure expansive
• Chain recurrent set
• Homoclinic class
• Generic
• Hyperbolic