Dimension of planar non-conformal attractors with triangular derivative matrices

TL;DR

This paper investigates the dimension of non-conformal, non-affine attractors generated by parametrized iterated function systems, establishing conditions under which their dimensions can be precisely determined using pressure and Lyapunov exponents.

Contribution

It introduces a transversality condition that allows exact dimension calculations for a broad class of non-conformal attractors with triangular derivative matrices.

Findings

Dimensions equal to the root of the subadditive pressure for almost all parameters.

Lyapunov dimension matches the attractor's dimension under the transversality condition.

Concrete examples satisfying the transversality condition are provided.

Abstract

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.