Problem Sheet F

This Problem Sheet is based on Lectures 11 and 12.

• Problem 1 asks you to compute the topological entropy of the shift map.
• Problem 2 asks you to compute the ball dimension of two somewhat exotic spaces.
• Problem 3 constructs an example of a dynamical system on the interval with infinite topological entropy.
• Problem 4 shows that transitive dynamical systems on the interval are always surjective and always have at least one interior fixed point.
• Problem 5 gives another criteria for a transitive dynamical system on the interval to be mixing. This problem is quite hard.