Speaker
Zhiyuan Zhang, Imperial College London
Abstract
Let \(\{f_i(x)=M_ix+a_i\}_{i\in I}\) be an affine iterated function system on \(\mathbb{R}^d\) with attractor \(\Lambda\), and let \(\mu\) be an associated self-affine measure. Assume that the \(M_i\) generate a Zariski dense subgroup of \({\rm GL}_d(\mathbb{R})\). Under the strong open set condition, we show that \(\dim_{H} \Lambda\) equals the affinity dimension and that \(\dim_{H} \mu\) equals the Lyapunov dimension.
The above results are consequences of the following projection theorem, which requires no separation condition. For every \(1\leq n\leq d\) and for Furstenberg measure a.e. \((d-n)\)-dimensional subspace \(V\),
\(\dim_{H} (P^V)_*\mu=\min\{n, \dim_{H} \mu\}.\)
This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures,
exploited through Wu’s one-dimensional entropy increase theorem. A key ingredient in reducing the
higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.
This is a joint work with Jialun Li and Wenyu Pan.