Continuum-wise expansive homoclinic classes for robust dynamical systems

Abstract

In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of C 1 -robustly expansive dynamical system. First, we show that if the homoclinic class 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) does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic.