Problem Sheet B

This Problem Sheet is based on Lectures 3 and 4.

• Problem 1 gives an alternative way to characterise the non-wandering set.
• Problem 2 shows that the chain recurrent set is independent of the metric when the space is compact.
• Problem 3 asks you to show that tent map has sensitive dependence on initial conditions.
• Problem 4 asks you to prove that the doubling map is a factor of the shift map.
• Problem 5 shows that sensitive dependence on initial conditions is not a dynamical invariant on non-compact metric spaces.
• Problem 6 gives another way to define chaos on metric spaces without isolated points.
• Problem 7 is about products of rotations on the torus. This problem is quite hard. 😈