Reference no: EM13882925

Problem 1: Let Qc(x) = x2 + c. Prove that if c <1/4, there is a unique µ > 1 such that Qcis topologically conjugate to Fµ (x) = µx (1 - x) via a map of the form h (x) = αx + β.

Problem 2: A point p is a non-wandering point for f, if , for any open interval J containing p, there exists x ε J and n > 0 such that fn (x) ε 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 closet set.

b. If Fµ is the quadratic map with µ > 2 + √5, show that Ω(Fµ) = A.

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