Just question 1.7.

1.6. Define the map f(x) = 2x (mod 1) on the unit interval [0, 1]. Let L denote the subinterval [0, 1/2] and R the subinterval [1/2, 1]. (a) Draw a chart of the itineraries of f as in Figure 1.12. (b) Draw the transition graph for f. (c) Establish sensitive dependence for orbits under this map. Show that each point has neighbors arbitrarily near that eventually map at least 1/2 unit apart. 1.7. Define the tent map on the unit inverval [0, 1] by [ 2x T( x) = if 0 S x $ 1/2 2(1 - x) if1/2