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 .
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.
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.