The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2

Abstract

We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension $$d-\frac{3}{2} $$ [in the $$(d-1)$$ -dimension space of d-IETs] for $$d\ge 5$$ . For $$d=4$$ this was shown by Athreya–Chaika and for $$d\in \{2,3\}$$ the set is known to have dimension $$d-2$$ . This provides lower bounds on the Hausdorff dimension of non-weakly mixing IETs and, with input from Al-Saqban et al. (Exceptional directions for the Teichmuller geodesic flow and Hausdorff dimension, 2017. arXiv:1711.10542 ), identifies the Hausdorff dimension of non-weakly mixing IETs with permutation $$(d,d-1,\ldots ,2,1)$$ when d is odd.