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A Family of Circle Maps

We denote by tex2html_wrap_inline522 the circle homeomorphism induced by tex2html_wrap_inline466 . We choose a reference direction in tex2html_wrap_inline526 , and parameterize tex2html_wrap_inline410 by the directions tex2html_wrap_inline530 of the supporting rays. With this parameterization, tex2html_wrap_inline408 becomes a family of Lipschitz homeomorphisms of the standard circle tex2html_wrap_inline534 . The rotation number tex2html_wrap_inline412 is a continuous, nondecreasing function. This function tex2html_wrap_inline504 contains significant information about the family tex2html_wrap_inline540 of twist maps.



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

tex2html_wrap606 Fig. 7. Proof of the formula tex2html_wrap_inline558 .
tex2html_wrap608 Fig. 8. Proof of the formula displaymath560
Recall that tex2html_wrap_inline570 is a point of increase for a nondecreasing function, g, if tex2html_wrap_inline574 for all sufficiently small tex2html_wrap_inline576 . Denote with tex2html_wrap_inline578 the standard rotation by tex2html_wrap_inline580 .


Remark. Case 2. in this proposition, when tex2html_wrap_inline604 is especially important.

Oliver Knill, Wed Jul 8, 1998