Repelling hyperbolic Cantor set of a bimodal map

In this article, we discuss the dynamical properties of a one-parameter family of symmetric bimodal maps Bα: x ↦ (1 - α)x + αx3, such as period doubling bifurcation, basin of attraction, Liapunov exponent and topological entropy, etc. Further, we show that for α > 4, there exists a tripling Cantor set Cα which is generated by Bα. Also, we prove that the topological entropy of Bα restricted to Cα is always positive. Finally, we prove the existence of a repelling hyperbolic Cantor set associated with a symmetric bimodal map.