Hausdorff dimension of planar self-affine sets and measures

Abstract

Let \(X={\bigcup }{\varphi }_{i}X\) be a strongly separated self-affine set in \({\mathbb {R}}^2\) (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the \(\varphi _{i}\), we prove that \(\dim X\) is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.

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Fig. 1
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Notes

  1. 1.

    The non-compactness assumption just means that not all of the maps \(\varphi _i\) are similarities. Under this assumption, total irreducibility is equivalent to the property that no line, or union of two lines, is invariant under all of the \(A_i\). We can relax the latter slightly, see 6.3.

  2. 2.

    Our methods actually apply to the images under the standard symbolic coding map of quasi-product measures, i.e. those which are equivalent, up to a shift, on every cylinder set, and with Radon–Nykodim derivative bounded away from 0 and infinity. This is because in all entropy computations in the proofs, measures which are equivalent in this sense differ in entropy (w.r.t. any partition) by an addivie constant depending only on the bounds on the RN-derivatives; so one can just fix a representative and work with it. When the matrix parts of the IFS preserve a (multi) cone, the Kaenmaki measure is known to be quasi-Bernoulli. Thus, in this case, our methods give an invariant measure of maximal dimension. We thank Pablo Shmerkin for this remark.

  3. 3.

    This assumption just means that the attractor is not a single point, and also implies that \(\mu \) has positive dimension and has no atoms. Note that although SOSC rules out a common fixed point for \(\Phi \), exponential separation by itself does not: It might be that all the maps are linear, with common fixed point 0, but that exponential separation holds in the linear group for products of the matrices and, therefore, also for the associated affine maps. This assumption will only be used in Sect. 3.2, where it is needed to ensure continuity of 1-dimensional linear images of \(\mu \).

  4. 4.

    In the triangular case of Proposition 6.6, the statement below holds for every \(\psi \in {\mathbb {A}}_{2,1}\) whose kernel is bounded away (in \({\mathbb {RP}^1}\)) from the vertical direction. In particular, it holds \(\eta ^*\)-a.e., because, under the assumptions of the proposition, \(\eta ^*\) is supported on a compact set in the complement of the vertical direction. The results throughout the present section hold in the triangular case, under similar adjustment of the assumptions.

  5. 5.

    Theorem 2.8 of [17] contains a slight error. The condition \(\frac{1}{m}H(\nu ,{\mathcal {D}}_n)>\varepsilon \) there should be replaced by \(>2\varepsilon \), or by \(>c\varepsilon \)., where c is any constant larger than one, but then the dependence of \(\delta \) and the other parameters on \(\varepsilon \) depend on c. Also, the theorem is stated for dyadic partitions rather than q-adic, but this modification is harmless.

  6. 6.

    If \(\mu \) is generated by a system of triangular matrices as in Proposition 6.6, we must assume that \(\theta \) is supported on maps whose kernel is \(\epsilon \)-far (in \({\mathbb {RP}^1}\)) from the vertical direction.

  7. 7.

    These assumptions are used only to ensure that \(\eta ^*\) is well defined and has positive dimension, so the same proof works when the defining affine maps are triangular and satisfy the assumptions in Proposition 6.6.

  8. 8.

    In the triangular case of Proposition 6.6, the kernels of compositions of the maps in \(\Phi \) are bounded away from the vertical direction, so \(\theta _n^V\) satisfies the hypothesis for the corresponding entropy growth theorem.

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Acknowledgements

This work was supported by ERC Grant 306494. B.B. acknowledges support from the Grants NKFI PD123970, OTKA K123782, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. M.H. supported by ISF Grant 1704 and NSF Grant DMS-1638352. A.R. acknowledges support from the Herchel Smith Fund at the University of Cambridge.

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Bárány, B., Hochman, M. & Rapaport, A. Hausdorff dimension of planar self-affine sets and measures. Invent. math. 216, 601–659 (2019). https://doi.org/10.1007/s00222-018-00849-y

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Mathematics Subject Classification

  • Primary 28A80
  • Secondary 37C45
  • 37F35