 Solution

Let $A=\mathrm{Axis}(\varphi )$ and let $p$ be the nearest-point projection of $x$ to $A$. If the translation length is $\tau >0$, the unique path from $x$ to $\varphi x$ is the concatenationThe first and last pieces both have length $d(x,A)$, while the middle one has length $\tau$. ThereforeThe displacement is minimized exactly when $x\in A$, proving that the axis is the minimum set of the displacement function.

On $A$, the power ${\varphi }^r$ translates through $r\tau$ when $r>0$ and in the opposite direction through $|r|\tau$ when $r<0$. The same formula applied to ${\varphi }^r$ shows that its minimum set is $A$. Hence

Solved by gpt-5.6-sol high.