Metrics on Hitchin Components from Hölder Distortion

We observe Thurston’s asymmetric metric on Teichmüller space may be expressed in terms of the Hölder regularity of boundary maps. We then associate $2$-dimensional stratified loci in ${\mathbb{R}\mathbb{P}}^{n-1}$ to ${\text{PSL}}_n(\mathbb{R})$ Hitchin representations with $n>3$. We prove that measuring the relative Hölder distortion of these loci gives asymmetric metrics on the Hitchin component ${\text{Hit}}_n(S)$ with complete and geometrically meaningful symmetrizations. These are the first known geometrically significant complete metrics on ${\text{Hit}}_n(S)$ for $n>3$.