Explicit construction of Hamilton's counterexample for Implicit Function Theorem in Frechet Space

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In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: thetamapstotheta + 2pi/k$ on a circle can be "pushed a little bit" so that the new diffeomorphism only has $0$ as a $k$-periodic point, but $f(pi/k)neq3pi/k$. My question is: is there any reference where Hamilton's construction was explicitly expressed? How can one "push" the rotation to obtain a diffeomorphism that only has $0$ as its $k$-periodic point?



Thank you in advance!




analysis inverse-function-theorem diffeomorphism




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  • Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
    – Paul Sinclair
    Jul 21 at 21:31











  • Thank you Paul. I think your construction should work for me.
    – Xuxu
    Jul 24 at 3:56














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0
down vote

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In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: thetamapstotheta + 2pi/k$ on a circle can be "pushed a little bit" so that the new diffeomorphism only has $0$ as a $k$-periodic point, but $f(pi/k)neq3pi/k$. My question is: is there any reference where Hamilton's construction was explicitly expressed? How can one "push" the rotation to obtain a diffeomorphism that only has $0$ as its $k$-periodic point?



Thank you in advance!




analysis inverse-function-theorem diffeomorphism




share|cite|improve this question



















  • Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
    – Paul Sinclair
    Jul 21 at 21:31











  • Thank you Paul. I think your construction should work for me.
    – Xuxu
    Jul 24 at 3:56












up vote
0
down vote

favorite









up vote
0
down vote

favorite











In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: thetamapstotheta + 2pi/k$ on a circle can be "pushed a little bit" so that the new diffeomorphism only has $0$ as a $k$-periodic point, but $f(pi/k)neq3pi/k$. My question is: is there any reference where Hamilton's construction was explicitly expressed? How can one "push" the rotation to obtain a diffeomorphism that only has $0$ as its $k$-periodic point?



Thank you in advance!




analysis inverse-function-theorem diffeomorphism




share|cite|improve this question











In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: thetamapstotheta + 2pi/k$ on a circle can be "pushed a little bit" so that the new diffeomorphism only has $0$ as a $k$-periodic point, but $f(pi/k)neq3pi/k$. My question is: is there any reference where Hamilton's construction was explicitly expressed? How can one "push" the rotation to obtain a diffeomorphism that only has $0$ as its $k$-periodic point?



Thank you in advance!





analysis inverse-function-theorem diffeomorphism





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 21 at 6:09









Xuxu

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  • Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
    – Paul Sinclair
    Jul 21 at 21:31











  • Thank you Paul. I think your construction should work for me.
    – Xuxu
    Jul 24 at 3:56
















  • Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
    – Paul Sinclair
    Jul 21 at 21:31











  • Thank you Paul. I think your construction should work for me.
    – Xuxu
    Jul 24 at 3:56















Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
– Paul Sinclair
Jul 21 at 21:31





Consider $g : theta mapsto theta + 2pi/k + epsilon(1 - cos theta)$ for small $epsilon$
– Paul Sinclair
Jul 21 at 21:31













Thank you Paul. I think your construction should work for me.
– Xuxu
Jul 24 at 3:56




Thank you Paul. I think your construction should work for me.
– Xuxu
Jul 24 at 3:56















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