** Discrete random walks on SOL **

Joint with J. Brieussel.

**1. Case of **

**Theorem 1 (Furstenberg (1963))** * Let be an absolutely continuous probability measure on . Then 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 be a Zariski dense countable subgroup of , . There exists a non-degenerate symmetric measure on such that the harmonic measure is singular. *

They conjecture that for any finitely supported probability distribution on , 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 by acting by a hyperbolic one parameter subgroup, like

Can be viewed as with left-invariant metric

From the law of large numbers, it follows that trajectories converge to points of the boundary of the hyperbolic plane .

** 2.2. Result **

**Theorem 3** * Let be a non-abelian countable subgroup in SOL. *

*
*
- There exists a probability distribution on such that the corresponding harmonic measure on the circle is singular and non atomic.
- 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 smooth.
- For all , there exists a finitely supported probability distribution on SOL such that tend to and whose corresponding harmonic measure on the circle is singular and non atomic.

* *

** 2.3. Proof **

We use Bernoulli convolutions. Let . Consider sums with uniformly chosen signs. If , the resulting distribution is singular. If , it is uniform. If , it depends. For instance, if is Pisot, the distribution is singular (Erdös 1939). This is the case encountered when the measure is supported on a lattice of SOL.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri PoincarĂ©, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/