Question 1:

The skinny baker map was defined in Chapter 4. It is a map of the unit square S in the plane which exhibits many typical properties of chaotic maps. See Figure 4.3 of Chapter 4 to recall the geometric behavior


The equations are written there, but we write them again in a slightly different form:


The middle third of the rectangles that remain are mapped out on each iteration. Define the invariant set A of B to be the points that lie in Bn(S) for every positive and negative integer n. The invariant set is a middle-third Cantor set of vertical lines. We call the set A invariant because it has the property that B-1(A) = A.

A) Find the area of Bn(S). Note that B is an area-contracting map, hence its name.

B) Find a general formula for (area-contracting) skinny baker maps, where the strips have width w instead of 1/3. Find the area contraction factor (per iteration) and the Lyapunov exponents.

Question 2:

(A) Find the exact intervals on the coordinate axes corresponding to the forward images of B (the skinny baker map), .S1S2S3S4 , and the backward images of B, S-3S-2S-1S0. (it is OK to draw the unit square and the corresponding labelled rectangles like in fgures 5.12 and 5.13 and specify the points on the x-axis or y-axis respectively).

(B) Determine the sequence S-3S-2S-1S0.S1S2S3S4 corresponding to the fi xed points of period 1 and 2 for B.

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