# Shift invariant metrics on the 1-dimensional torus

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?