The Hyperbolicity Of Logistic Map And Its Perturbation

In this paper, we considered the behavior of the logistic maps f_μ(x)=μx(1-x),x∈I=[0,1],μ∈(4,4+ε). We prove that the set K_μ={x∈I:f_μⁿ(x)∈I, (?)n≥0}isa hyperbolic set forεsufficiently small. We have an expanding behaviour away from a smallneighbourhood of the unique critical point. When the orbit return too close to the criticalpoint, we have a loss in the exponential increasing. But we try to compensate it with thefinite forward iterates. Moreover, if g_μis the C² perturbation family of Logistic map, thenthe set K_gμ={x∈I:g_μⁿ(x)∈I,(?)n≥0} is also hyperbolic.