Showing that the following functions are not topologically conjugated

Question

Let $f(x) = x + \frac{1}{2} sin \;x$ and $g_k(x) = x + \frac{1}{2} h_k(x),$ where $$h_k(x) = x - \frac{x^3}{3 !} + \frac{x^5}{5 !} + ... + (-1)^k \frac{x^{2k+1}}{(2k + 1)!}, \; k \geq 1.$$

Show that for every $k \ge 1$, $f$ and $g_k$ are not topologically conjugated.

My work: For every $k \ge 1$, $g_k$ has a finite number of fixed points. Since $f$ has an infinite number of fixed points, it follows that for every $k \geq 1$, $f$ is not topologically conjugate to $g_k$.

Is it correct this argument?

Thank you!

Answer 1

Yes, this is correct. Explicitly, since $h_k$ is a polynomial of degree $2k+1$, it has at most $2k+1$ zeroes, so $g_k(x)$ has at most $2k+1$ fixed points. Note that this argument shows that $f$ and $g_k$ are not conjugate by any bijection at all, not merely that they are not topologically conjugate.