Homework 1: Due April 12th, Monday, 2010
1. Prove that supnN (sin n) = 1. 2. Find the asymptotic frequency of 0 being the second leading digit of power of 3. 3. For a circle map f : S 1 S 1 that is homeomorphic and orientation preserving, let us denote it
Homework 2: Due April 19th, Monday, 2010
1. Let f : [a, b] [a, b] be dierentiable with |f (x)| 1 for all x [a, b]. Prove that there is no periodic points of period greater than 2. 2. Prove that a homeomorphism cannot have eventually periodic points (meani
Homework 3: Due April 26th, Monday, 2010
Consider the logistic function: F (x) = x(1 x). 1. Describe the bifurcation that occurs when = 1 and 1 and sketch the bifurcation diagrams. ( = 3 was done in class.) 2. Show that the following piecewise linear map
Homework 4: Due May 3rd, Monday, 2010
Consider the logistic function: F (x) = x(1 x). We have shown that 0 = (1), 1 = (10), 2 = (1011), 3 = (10111010), . are the sequences for the periodic xed points at the end of the period doubling bifurcations. (Here,
Homework 5: Due May 17th, Monday, 2010
1. (1) If a holomorphic map f : D D (where D is the unit disk) xes the origin and is not a rotation, prove that the successive images f n (z ) (here f n means the successive compositions f f . by n times) converge to
Homework 6: Due May 24th, Monday, 2010
1. Show that fc (z ) = z 2 + c has an attractive 2-cycle if c is inside the circle of radius 1/4 centered at -1. 2. The boundary scanning method is the following. For some large number n, if |f n (z )| remains within