Reference no: EM13589018

1. Let Q_C(x) = x² + c. Prove that if c < 1/4, there is a unique μ > 1 such that Q_c 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 F_μ. is the quadratic map with μ > 2 + √5, show that SI(F_μ) = Λ.

c. Identify Ω(Fμ) for each μ satisfying 0 < µ ≤ 3.