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arXiv · 2611.09066

Hausdorff dimension of planar box-like self-affine sets with rotations

Antti Käenmäki·R. Dániel Prokaj·Károly Simon·Sascha Troscheit
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Abstract

We study the Hausdorff dimension of proximal planar self-affine sets whose defining maps have diagonal or anti-diagonal linear parts, with both types present. Such systems are irreducible but not strongly irreducible. Under a projective exponential separation condition, we show that the Hausdorff dimension equals the minimum of $2$ and the affinity dimension, thereby extending the dimension formula of Hochman--Rapaport to all proximal irreducible planar systems. When the affinity dimension is at most an explicit threshold determined by the Lyapunov exponents, the formula holds assuming only that there are no exact overlaps. Above the threshold, it holds under the strong open set condition. These dimension formulae are proved by approximation with finite diagonal subsystems. Assuming projective exponential separation, we also determine the pointwise dimensions of the coordinate projections of the equilibrium state. Below the threshold, the projections are not exact-dimensional: the pointwise dimension of each exists almost everywhere and takes exactly two values, each on a set of positive measure. At or above the threshold, both projections are exact-dimensional with dimension one.

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Antti Käenmäki, R. Dániel Prokaj, Károly Simon, Sascha Troscheit. 2026-11-06. Hausdorff dimension of planar box-like self-affine sets with rotations. https://arxiv.org/abs/2611.09066

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