2
$\begingroup$

thanks in advance for helping me. First I'll introduce some definitions:

(1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving diffeomorphism of the circle. Then $f$ may be lifted to a homeomorphism $F:\mathbb{R} \rightarrow \mathbb{R}$ of the real line, satistying $$F(x+m)=F(x)+m.$$ The rotation number $\rho(f)$ of $f$ is defined as the fractional part of $$\rho_{0}(f) = \lim_{n\to\infty} \frac{F^{n}(x)}{n}.$$ That is, $\rho(f)$ is in the unique number in $[0,1)$ for which $\rho_{0}(f)-\rho(f)$ is an integer. The rotational number exists (proved by Poincaré) and is independent of the starting point $x$.

(2) An orientation of a smooth m-dimensional manifold $M$ is a choice of orientation on every tangent space $T_{x}M$ satisfying the condition: for each $x \in M$ there is a neighborhood $U \subset M$ and an orientation-preserving diffeomorphism (i.e. a coordinate system) $h$ from $U$ to an open set in $\mathbb{R}^{n}$

(3) Let $f$ and $g$ be two k-times differentiable functions from $\mathbb{R} $ to $\mathbb{R}$. The $C^{1}$-distance between $f$ and $g$ is given by $$d_{1}(f,g)=\sup_{x\in \mathbb{R}}(|f(x)-g(x)|,|f^{(1)}(x)-g^{(1)}(x)|).$$

Now, my question: how to prove the following proposition using Sard's theorem?

Suppose $f$ is an orientation-preserving diffeomorphism of the circle with $\rho(f) = 0$. Then there is a $C^{1}$ (meaning continuously differentiable) diffeomorphism $g$ which is arbitrarily close to $f$ with respect to the $C^{1}$-distance and which has only isolated fixed points.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.