Hausdorff dimension of the Rauzy gasket

Abstract

The Rauzy gasket is the attractor of a parabolic, nonconformal iterated function system on the projective plane which describes an exceptional parameter set in various important topological and dynamical problems. Since 2009 there have been several attempts to calculate the Hausdorff dimension of the Rauzy gasket, which is a challenging problem due to the combination of the parabolicity and nonconformality in the geometry. In this paper we settle this question by proving that the Hausdorff dimension of the Rauzy gasket equals the (projective) affinity dimension. The key technical result underpinning this is a partial generalisation of work of Hochman and Solomyak to the $\mathrm{SL}_3(\mathbb{R})$ setting, where we establish the exact dimension of stationary (Furstenberg) measures supported on the Rauzy gasket. The dimension results for both stationary measures and attractors are established in broader generality and extend recent work on projective iterated function systems to higher dimensions.