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Furstenberg boundary

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In potential theory, a discipline within applied mathematics, the Furstenberg boundary is a notion of boundary associated with a group. It is named for Harry Furstenberg, who introduced it in a series of papers beginning in 1963 (in the case of semisimple Lie groups). The Furstenberg boundary, roughly speaking, is a universal moduli space for the Poisson integral, expressing a harmonic function on a group in terms of its boundary values.

Motivation

A model for the Furstenberg boundary is the hyperbolic disc D = { z : | z | < 1 } {\displaystyle D=\{z:|z|<1\}} . The classical Poisson formula for a bounded harmonic function on the disc has the form

f ( z ) = 1 2 π ∫ 0 2 π f ^ ( e i θ ) P ( z , e i θ ) d θ {\displaystyle f(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }{\hat {f}}(e^{i\theta })P(z,e^{i\theta })\,d\theta }

where P is the Poisson kernel. Any function f on the disc determines a function on the group of Möbius transformations of the disc by setting F(g) = f(g(0)). Then the Poisson formula has the form

F ( g ) = ∫ | z | = 1 f ^ ( g z ) d m ( z ) {\displaystyle F(g)=\int _{|z|=1}{\hat {f}}(gz)\,dm(z)}

where m is the Haar measure on the boundary. This function is then harmonic in the sense that it satisfies the mean-value property with respect to a measure on the Möbius group induced from the usual Lebesgue measure of the disc, suitably normalized. The association of a bounded harmonic function to an (essentially) bounded function on the boundary is one-to-one.

Construction for semi-simple groups

In general, let G be a semi-simple Lie group and μ a probability measure on G that is absolutely continuous. A function f on G is μ-harmonic if it satisfies the mean value property with respect to the measure μ:

f ( g ) = ∫ G f ( g g ′ ) d μ ( g ′ ) {\displaystyle f(g)=\int _{G}f(gg')\,d\mu (g')}

There is then a compact space Π, with a G action and measure ν, such that any bounded harmonic function on G is given by

f ( g ) = ∫ Π f ^ ( g p ) d ν ( p ) {\displaystyle f(g)=\int _{\Pi }{\hat {f}}(gp)\,d\nu (p)}

for some bounded function f ^ {\displaystyle {\hat {f}}} on Π.

The space Π and measure ν depend on the measure μ (and so, what precisely constitutes a harmonic function). However, it turns out that although there are many possibilities for the measure ν (which always depends genuinely on μ), there are only a finite number of spaces Π (up to isomorphism): these are homogeneous spaces of G that are quotients of G by some parabolic subgroup, which can be described completely in terms of root data and a given Iwasawa decomposition. Moreover, there is a maximal such space, with quotient maps going down to all of the other spaces, that is called the Furstenberg boundary.

References

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