a significant progress in dimension theory of affine IFSs, showing that the Hausdorff and affinity dimensions of a planar self-affine set coincide under the strong separation condition and certain irreducibility assumption; and similarly, the Hausdorff and Lyapunov dimensions of a planar self-affine measure coincide under the same assumptions.

Organization of the paper
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The paper is organized as follows. In Section \[S-2\], we provide some density results about conditional measures, and present a version of Oseledets’s multiplicative ergodic theorem due to Froyland et al. [@FroylandLloydQuas2010]. In Section \[S-3\], we give some auxiliary results on the coding maps for average contracting affine IFSs. In Section \[S-4\], we construct a finite family of measurable partitions of $\Sigma$ for a given average contracting affine IFS and give some necessary properties. In Section \[S-5\] we prove an inequality for the transverse dimensions of the conditional measures associated with these measurable partitions. In Section \[S-6\], we prove Theorems \[thm-1.0\]-\[thm-1.2\], \[cor-1.0\]-\[thm-1.7’\]. In Section \[S-7\], we give some properties of Lyapunov dimension. In Section \[S-8\], we prove Theorems \[thm-1.3\]-\[thm-1.4\].

Preliminaries {#S-2}
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In this section, we first give the definitions and some properties of conditional entropies and conditional measures. Then we collect some known facts about induced 