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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
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Recall that 
is a point of increase
for a nondecreasing function, g,
if 
for all sufficiently small
. Denote with 
the standard rotation by
.
Remark. Case 2. in this proposition, when 
is especially important.
Oliver Knill, Wed Jul 8, 1998
