### Dimension of planar self-affine measures

Ergodic Theory and Dynamical Systems Seminar

6th June 2019, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

'A compactly supported Borel probability measure $\mu$ on

$\mathbb{R}^{2}$ is called self-affine if there exist affine contractions

$\varphi_{1},...,\varphi_{m}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$

and a probability vector $(p_{i})_{i=1}^{m}$ such that $\mu=\sum_{i=1}^{m}p_{i}\cdot\mu\circ\varphi_{i}^{-1}$.

It is well known that there exists a value $\dim_{L}\mu$, called

the Lyapunov dimension, which is the ''expected'' value for the dimension

of $\mu$. I will discuss a recent project with Mike Hochman, building

on our joint work with Balázs Bárány, in which we establish the equality

$\dim\mu=\dim_{L}\mu$ under mild assumptions.

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