1. Let QC(x) = x2 + c. Prove that if c < 1/4, there is a unique μ > 1 such that Qc is topologically conjugate to Fμ(x) = μx(1 - x) via a map of the form h(x) = αx+β.
2. A point p is a non-wandering point for f, if, for any open interval J containing p, there exists x E J and n > 0 such that fn(x) E J. Note that we do not require that p itself return to J. Let Ω(f) denote the set of non-wandering points for f.
a. Prove that Ω(f) is a closed set.
b. If Fμ. is the quadratic map with μ > 2 + √5, show that SI(Fμ) = Λ.
c. Identify Ω(Fμ) for each μ satisfying 0 < µ ≤ 3.