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.

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About metric2011

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