Circle map

Remarks

Example: for $K = 0$ the circle map is the rotation $x \mapsto x + \Omega$; for irrational $\Omega$ it is an irrational rotation.

The derivative is $F'_{\Omega, K}(x) = 1 - K \cos(2\pi x)$. For $0 \le K < 1$ it is positive everywhere, so $f_{\Omega, K}$ is a diffeomorphism of the circle. For $K = 1$ the derivative vanishes at $x = 0$, so $f_{\Omega, 1}$ is a homeomorphism with a critical point, but not a diffeomorphism.

Non-example: for $K > 1$ the derivative takes negative values, so $F_{\Omega, K}$ is not monotone and $f_{\Omega, K}$ is not invertible. The map is still defined, but it is not a homeomorphism.

The parameter $\Omega$ is often taken in $[0, 1)$, since $f_{\Omega + 1, K} = f_{\Omega, K}$.