Let $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ be the 1-dimensional torus with the usual topology. It is well known that this is a metric space with the distance $d(x+\mathbb{Z},y+\mathbb{Z}):=\min_{n\in \mathbb{Z}}|x-y+n|$. Clearly this metric is shift-invariant, in the sense that $d(x+z+\mathbb{Z},y+z+\mathbb{Z})=d(x+\mathbb{Z},y+\mathbb{Z})$ for any $z\in \mathbb{R}$. Moreover, for any $\alpha\in(0,1]$ we have that $d(x,y)^\alpha$ is also a metric on $\mathbb{T}$ which generates the same topology and these metrics are also shift-invariant. These metrics are not (strongly) equivalent in the sense that in general there are no absolute constants $c,c'>0$ such that for all $x,y\in \mathbb{T}$, $cd(x,y)^\alpha\le d(x,y)^\beta\le c'd(x,y)^\alpha$ for $\alpha\not=\beta$.

My question: Is there any classification of shift-invariant metrics which generate the same (usual) topology on $\mathbb{T}$ up to (strong) equivalence?



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