Solving the Babylonian problem of quasiperiodic rotation rates

A trajectory $u_n := F^n(u_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $d$-dimensional torus, and there is a choice of coordinates on the torus $\mathbb{T}$ for which $F$ has the form $F(\theta) = \theta + \rho\bmod1$ for all $\theta\in\mathbb{T}$ and for some $\rho\in\mathbb{T}$. There is an ancient literature on computing three rotation rates $\rho$ for the Moon. %There is a literature on determining the coordinates of the vector $\rho$, called the rotation rates of $F$. (For $d>1$ we always interpret $\bmod1$ as being applied to each coordinate.) However, even in the case $d=1$ there has been no general method for computing $\rho$ given only the trajectory $u_n$, though there is a literature dealing with special cases. Here we present our Embedding Continuation Method for computing some components of $\rho$ from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. Rotation rates are often called "rotation numbers" and both refer to a rate of rotation of a circle. However, the coordinates of $\rho$ depend on the choice of coordinates of $\mathbb{T}$. We explore the various sets of possible rotation rates that $\rho$ can yield. We illustrate our ideas with examples in dimensions $d=1$ and $2$.