## Notes of Ryokichi Tanaka’s lecture

Discrete random walks on SOL

Joint with J. Brieussel.

1. Case of ${SL(d,{\mathbb R})}$

Theorem 1 (Furstenberg (1963)) Let ${\mu}$ be an absolutely continuous probability measure on ${SL(2,{\mathbb R})}$. Then ${W_n.x}$ converges to the circle, with asymptotic distribution equal to harmonic measure.

This is not always the case. For instance,

Theorem 2 (Kaimanovitch, Le Prince 2011) Let ${\Gamma}$ be a Zariski dense countable subgroup of ${SL(d,{\mathbb R})}$, ${d\geq 2}$. There exists a non-degenerate symmetric measure ${\mu}$ on ${\Gamma}$ such that the harmonic measure is singular.

They conjecture that for any finitely supported probability distribution ${\mu}$ on ${SL(d,{\mathbb R})}$, the corresponding harmonic measure on the flag manifold is singular. But there is a counterexample, due to Barany-Pollicott-Simon (2012). It is non symmetric. Bourgain improved it into a symmetric example.

2. Case of SOL

2.1. What is SOL ?

This is the semi-direct product of ${{\mathbb R}^2}$ by ${{\mathbb R}}$ acting by a hyperbolic one parameter subgroup, like

$\displaystyle \begin{array}{rcl} \begin{pmatrix} e^z & 0 \\ 0 & e^{-z} \end{pmatrix}. \end{array}$

Can be viewed as ${{\mathbb R}^3}$ with left-invariant metric

$\displaystyle \begin{array}{rcl} dz^2+e^{2z}dx^2+e^{-2z}dy^2. \end{array}$

From the law of large numbers, it follows that trajectories converge to points of the boundary of the hyperbolic plane ${\{y=0\}}$.

2.2. Result

Theorem 3 Let ${\Gamma}$ be a non-abelian countable subgroup in SOL.

1. There exists a probability distribution ${\mu}$ on ${\Gamma}$ such that the corresponding harmonic measure on the circle is singular and non atomic.
2. On the other hand, there exists a finitely supported probability distribution on SOL whose harmonic measure is absolutely continuous. Moreover, one can assume that the density is ${C^k}$ smooth.
3. For all ${\alpha>0}$, there exists a finitely supported probability distribution on SOL such that ${S_n/n}$ tend to ${\alpha}$ and whose corresponding harmonic measure on the circle is singular and non atomic.

2.3. Proof

We use Bernoulli convolutions. Let ${0<\lambda<1}$. Consider sums ${\sum\pm \lambda^n}$ with uniformly chosen signs. If ${\lambda<\frac{1}{2}}$, the resulting distribution is singular. If ${\lambda=\frac{1}{2}}$, it is uniform. If ${\lambda>\frac{1}{2}}$, it depends. For instance, if ${\lambda^{-1}}$ is Pisot, the distribution is singular (Erdös 1939). This is the case encountered when the measure is supported on a lattice of SOL.