Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

Abstract

The sub-additive pressure function P (s) for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution s 0 to P (s 0 ) = 1, were introduced by K. Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affinities which define the IFS are 2 × 2 matrices which strictly preserve a common cone, the sub-additive pressure is locally real analytic as a function of the matrix coefficients of the linear parts of the affinities. In this setting we also show that the sub-additive pressure is piecewise real analytic in s, implying that the affinity dimension is locally analytic in the matrix coefficients. Combining this with a recent result of Bárány, Hochman and Rapaport we obtain results concerning the analyticity of the Hausdorff dimension for certain families of planar self-affine sets.

Typical absolute continuity for classes of dynamically defined measures

“…Furthermore, one could study the absolute continuity of the Furstenberg measure induced by the Käenmäki measure (that is, the natural equilibrium measure for self-affine IFS, see [21]). For self-affine systems whose linear parts are strictly positive matrices the Käenmäki measure is a Gibbs measure which smoothly depends on the matrix elements, see Bárány and Rams [7] and Jurga and Morris [20]. The absolute continuity and the dimension of the Furstenberg measure induced by the Käenmäki measure plays a central role in the calculation of the dimension of the Käenmäki measure, see [7].…”