On self-affine measures associated to strongly irreducible and proximal systems.(English)Zbl 07863031
Summary: Let \(\mu\) be a self-affine measure on \(\mathbb{R}^d\) associated to an affine IFS \(\Phi\) and a positive probability vector \(p\). Suppose that the maps in \(\Phi\) do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that \(\dim \mu\) is equal to the Lyapunov dimension \(\dim_L (\Phi, p)\) whenever \(d = 3\) and \(\Phi\) satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring \(\dim \mu = \min \{d, \dim_L (\Phi, p)\}\), from which earlier results in the planar case also follow. Additionally, we prove that \(\dim \mu = d\) whenever \(\Phi\) is Diophantine (which holds e.g. when \(\Phi\) is defined by algebraic parameters) and the entropy of the random walk generated by \(\Phi\) and \(p\) is at least \((\chi_1 - \chi_d) \frac{(d - 1) (d - 2)}{2} - \sum_{k = 1}^d \chi_k\), where \(0 > \chi_1 \geq \ldots \geq \chi_d\) are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of \(\mu\).
Keywords:
self-affine measure;dimension of measures;Lyapunov dimension;strong irreducibility;proximalityCite
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