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It is well known that for $a>4$, the dynamical behaviors of the logistic map$f_a(x)=ax(1-x)$ on the maximal invariant compact set are "simple" which couldbe clearly explained by the theories of hyperbolic dynamics and symbolicdynamics. Is it possible that similar phenomena could be observed in generalreal polynomial maps? In this paper, we study this problem by investigating thereal polynomial map $f_a(x)=ag(x)$, where $a$ is a parameter, and $g$ is areal-coefficient polynomial, which has at least two different real zeros oronly one real zero.

Hyperbolicity of the invariant sets for the real polynomial maps