Fibre stability for dominated self-affine sets

Abstract

Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\underset{E\in \mathrm{Tan}(K)}{\max }\underset{x\in \ddot{I}{€}_{V^{\perp }}(E)}{\max }\mathrm{di}{\mathrm{m}}_{\mathrm{H}}(\ddot{I}{€}_{V^{\perp }}^{-1}(x)\cap E)=\mathrm{di}{\mathrm{m}}_{\mathrm{A}}K-\mathrm{di}{\mathrm{m}}_{\mathrm{A}}\ddot{I}{€}_{V^{\perp }}(K).$$ Here, $\mathrm{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\underset{x\in \ddot{I}{€}_{V^{\perp }}(K)}{\max }\mathrm{di}{\mathrm{m}}_{\mathrm{H}}(\ddot{I}{€}_{V^{\perp }}^{-1}(x)\cap K)=\mathrm{di}{\mathrm{m}}_{\mathrm{A}}K-\mathrm{di}{\mathrm{m}}_{\mathrm{A}}\ddot{I}{€}_{V^{\perp }}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.

35 pages + 3 page appendix, 1 figure