The geometry of Cantor systems associated with an asymptotically non-hyperbolic family $(f_\epsilon)_{0\le\epsilon\le \epsilon_0}$ was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc.351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of $f_\epsilon$ behaves like $1- K\epsilon^{1/\gamma}$ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Neumann transformations. PhD Thesis, CUNY Graduate Center, May 1999).