## |

We denote by the
circle homeomorphism induced by
. We choose a reference direction in , and parameterize
by the directions of the supporting rays.
With this parameterization, becomes a
family of Lipschitz homeomorphisms of the *standard circle*
. The rotation number is a
continuous, nondecreasing function. This function
contains significant information about the family
of twist maps.

Proof. See Fig. 7. and Fig. 8.

Fig. 7. Proof of the formula . |

Fig. 8. Proof of the formula |

Remark. Case 2. in this proposition, when is especially important.