Let $\gamma:[a,b]\longrightarrow\mathbb{C}$ be a closed continuous curve in the complex plane satisfies:


$(1)\ \gamma(t)\neq 0,\ \forall t\in [a,b]$;


$(2)\ \{\frac{\gamma(t)}{|\gamma(t)|}:t\in [a,b]\}=\{z\in \mathbb{C}:|z|=1\}$.


For any given positive integer $n$, is that right that we can find $a\leq t\_1<t\_2\leq b$ such that $\gamma(t\_1)^n=\gamma(t\_2)^n?$



A question about a closed continuous curve in the complex plane
If $\gamma$ has non-zero winding number, then Nate Eldridge's comment shows that the answer is positive. However, note that for (for example) $n = 2,$ the logarithmic spiral (which is not a closed curve) is a counterexample To make it closed, take the boundary of a thin tubular neighborhood of a long piece of said spiral.


