Rotation number

Remarks

Example: the rotation $R_\alpha(x) = x + \alpha$ of $S^1$ has the lift $F(x) = x + \alpha$, and $F^n(x) - x = n\alpha$. So $\rho(F) = \alpha$, and $\rho(R_\alpha)$ is the fractional part of $\alpha$.

Non-example: the doubling map $x \mapsto 2x$ of $S^1$ is not a homeomorphism. Its lift $F(x) = 2x$ satisfies $F(x + 1) = F(x) + 2$, and $(F^n(x) - x)/n = (2^n - 1) x / n$ has no finite limit for $x \neq 0$, so no rotation number is defined.

The rotation number is invariant under conjugacy: if $h$ is an orientation-preserving homeomorphism of $S^1$, then $\rho(h^{-1} \circ f \circ h) = \rho(f)$. It depends continuously on $f$ in the uniform topology. The fractional part is often left out, and $\rho(f)$ is viewed as an element of $\R / \Z$.