In this lecture we restrict our attention to orientation-preserving reversible dynamical systems on $S^1$. All such systems have topological entropy zero, but this does not mean that they are dynamically uninteresting.

We associate to such a system $f$ its rotation number $\operatorname{rot}(f) \in S^1$.  The main properties of the rotation number are:

  • The rotation number of a circle rotation is given by (surprise!) $\operatorname{rot}( \rho_{ \theta}) = \theta$.
  • If $ \operatorname{rot}(f)$ is rational, the dynamics are simple. Periodic points exist, and $f$ is not transitive.
  • If $ \operatorname{rot}(f)$ is irrational, the dynamics are more complicated: either all orbits are dense or all orbits as asymptotic to a Cantor set.

We only address the first bullet point today. The second one will come next lecture, and the third next week.

Comments and questions?