This Problem Sheet is based on Lectures 13 and 14.

  • Problem 1 asks you to show that transitive maps always have a periodic point of period 6. Remark: Recall 6 is the largest even number in the Sharkovsky ordering.
  • Problem 2 gives an alternative characterisation of turbulence.
  • Problem 3 is concerned with dynamical systems on intervals that fix an endpoint.
  • Problem 4 is about the graph of a periodic orbit.
  • Problem 5 introduces the notion of $k$-turbulence. This problem implies that if $f \colon [0,1] \to [0,1]$ is turbulent then $f^n$ is $2^n$-turbulent, and hence also turbulent. The last part of the problem generalises Theorem 13.17.

This Problem Sheet is (hopefully) easier than the last one. (This is to make up for the fact that the lectures this week were more difficult than normal.)

Comments and questions?