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Math Help - Dynamics in S^1: rotation number

  1. #1
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    Dynamics in S^1: rotation number

    Suppose f,g in Hom_+(S^1),i.e, orientation-preserving homeomorphisms of the circle (deg(f) = deg(g) = 1), such that f o g = g o f

    Show that rho(f o g) = rho(f) + rho(g) (mod 1), where rho is the rotation number.


    This is where I got so far:


    Let F and G be liftings to R of f and g, respectively.



    Then F o G and G o F are both liftings of f o g = g o f.
    So there is an integer k such that F o G = G o F + k


    Then resolving the recurrence a_{n+1} = a_n + n,
    one can check that (F o G)^n = F^n o G^n -(n(n-1)/2).k (I)


    Then ((F o G)^n(x)-x)/n = (F^n(G^n(x)) - G^n(x))/n + (G^n(x)-x)/n - ((n-1)/2)k
    The problem here is that when taking limit in n->inf, the last term of the right side of the equation goes to -inf, which doesnt make sense, since Poincare says that the limit in n of all the other terms in that equation are finite


    I probably messed up in calculating (I), but I believe that(n(n-1)/2) is the number of permutations required for getting from F(G(F(G(...(F(G(F(G(x)))))...)))) to F(F(F(F(...(G(G(G(G((x))))))...)))) and in each one of those, we permute G o F by F o G, thus a -k appearing in each step.


    I guess if I find liftings which conmmute, then the problem is solved. But I dont see why that should happen



    Well, thanks in advance
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  2. #2
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    Using commutativity and the fact that the rotation number is independent of x, we get:

    <br />
\[<br />
\begin{array}{ccl}<br />
\rho (f\circ g)&=&\displaystyle{\lim_{n\rightarrow \infty} \frac{(F\circ G)^n(x)-x}{n}}\\\\<br />
&=&\displaystyle{\lim_{n\rightarrow \infty} \frac{F^n(G^n(x))-x}{n}}\\\\<br />
&=&\displaystyle{lim_{n\rightarrow \infty} \frac{F^n(y)-y}{n}}-\displaystyle{\lim_{n\rightarrow \infty}\frac{G^{-n}(y)-y}{n}}\\\\<br />
&=&\rho(f)-\rho(g^{-1})\> \text{mod} \>1\\\\<br />
&=&\rho(f)+\rho(g)\> \text{mod} \>1<br />
\end{array}<br />
\]<br />
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