Hausdorff dimension of planar self-affine sets and measures

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with Barany to the overlapping case.

[1]  K. Falconer,et al.  Planar self-affine sets with equal Hausdorff, box and affinity dimensions , 2015, Ergodic Theory and Dynamical Systems.

[2]  Ariel Rapaport.  On self-affine measures with equal Hausdorff and Lyapunov dimensions , 2015, 1511.06893.

[3]  On the dimension of Bernoulli convolutions for all transcendental parameters , 2018, Annals of Mathematics.

[4]  T. Bedford.  The box dimension of self-affine graphs and repellers , 1989 .

[5]  B. Solomyak.  Measure and dimension for some fractal families , 1998, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[7]  De-Jun Feng,et al.  Dimension theory of iterated function systems , 2009, 1002.2036.

[8]  De-Jun Feng,et al.  Self-affine sets in analytic curves and algebraic surfaces , 2016, 1612.07136.

[9]  K. Simon,et al.  Hausdorff Dimension for Randomly Perturbed Self Affine Attractors , 2007 .

[10]  Balázs Bárány,et al.  Dimension of self‐affine sets for fixed translation vectors , 2016, J. Lond. Math. Soc..

[11]  K. Lau,et al.  Relationships between Different Dimensions of a Measure , 2000 .

[12]  M. Hochman,et al.  Hausdorff dimension of planar self-affine sets and measures , 2017, Inventiones mathematicae.

[13]  B. Bárány.  On the Ledrappier–Young formula for self-affine measures , 2015, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  I. Kocyigit,et al.  On the Dimension of Self-Affine Fractals , 2013 .

[15]  De-Jun Feng.  Dimension of invariant measures for affine iterated function systems , 2019, Duke Mathematical Journal.

[16]  Steven P. Lalley,et al.  Falconer's formula for the Hausdorff dimension of a self-affine set in R2 , 1995, Ergodic Theory and Dynamical Systems.

[17]  A. Kaenmaki,et al.  Ledrappier-Young formula and exact dimensionality of self-affine measures , 2015, 1511.05792.

[18]  Steven P. Lalley,et al.  Hausdorff and box dimensions of certain self-affine fractals , 1992 .

[19]  Ville Suomala,et al.  Existence of doubling measures via generalised nested cubes , 2010, 1011.0683.

[20]  Lai-Sang Young,et al.  Ergodic Theory of Differentiable Dynamical Systems , 1995 .

[21]  Curtis T. McMullen,et al.  The Hausdorff dimension of general Sierpiński carpets , 1984, Nagoya Mathematical Journal.

[22]  I. Morris,et al.  On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems , 2016, Transactions of the American Mathematical Society.

[23]  Kenneth Falconer,et al.  The Hausdorff dimension of self-affine fractals , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  Michael Hochman,et al.  On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$ , 2012, 1503.09043.