Ledrappier-Young formula and exact dimensionality of self-affine measures

By Balázs Bárány, A. Kaenmaki. 2015

61 citations recorded by the source corpus.

In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier–Young formula.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e920b22f889bd966287c6d0770bf102ddb121bbc

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Citations in the corpus

Assouad Dimension and Fractal Geometry (2020, methodology): “Other more recent developments connecting the ergodic theory of the action of the collection of matrices in the defining IFS on projective space also play a vital role, see [17, 18, 19, 79].”
Dimension of invariant measures for affine iterated function systems (2019, methodology): “For a contracting invertible affine IFS, Rapaport [59] proved the implication (b)⇒ (1.11) in the case when m is Bernoulli and (Mj)j∈Λ satisfies an irreducibility assumption; whilst Bárány and Käenmäki [3] proved (1.11) under the assumptions that the conditions (1.13)-(1.14) hold for all i ∈ {1, ”
On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems (2016, background): “Very recently, Bárány and Käenmäki [3] proved that every self-affine measure on the plane is exact dimensional, and its exact dimension can be expressed in terms of the so-called Ledrappier-Young formula.”
Hausdorff dimension of planar self-affine sets and measures (2017, methodology): “A central tool in this theory is the Ledrappier-Young formula, which in the setting of self-affine measures is due to Käenmäki and Báráni [4, 12], and which we now recall (see also Section 2.”