Re: circle homeomorphism


"David C. Ullrich" <ullrich@xxxxxxxxxxxxxxxx> wrote in message
news:ft65l1tc7gd90vvjt159t7a12ebo5fi7ea@xxxxxxxxxx
> On Sun, 16 Oct 2005 11:46:21 +0100, "Thomas Novascott"
> <Thomas.novascott@xxxxxxxxxxxxxxxxxx> wrote:
>
>>> It seems we have an increasing homeomorphism F of R,
>>> with period 2\pi, such that
>>>
>>> f(eit(t)) = eit(F(t))
>>>
>>> (writing eit(t) = exp(2 Pi i t)); F is a "lift"
>>> of f (ie a lifting of f to R wrt the covering
>>> map eit:R -> S^1.)
>>>
>>> and the rotation number is the limit of
>>>
>>> (*) (F^n(t) - t)/n,
>>>
>>> (where F^n denotes iterated composition).
>>>
>>> It follows that
>>>
>>> eit(F(F(t)) = f(eit(F(t)) = f(f(eit(t));
>>>
>>> that is, F^2 is a lift of f^2. If we're willing
>>> to believe that the limit (*) exists the result
>>> you ask about follows easily...
Thank you for your indepth explanation,
i think i should have made it clearer, but i dont see how
eit(F(F(t)) = f(eit(F(t)) = f(f(eit(t));

proves p(f^n)=n.p(f)
(sorry for not making this clearer)

Thanks once again for your explanation


.

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