Conformal measure

The original definition of a conformal measure (or density) is due to S.J. Patterson [a2] in the case of a Fuchsian group $ \Gamma $ acting on the hyperbolic space $ B ^ {2} $( cf. Poincaré model). Its definition carries over to any dimension (D. Sullivan, [a4]):

Let $ \Gamma $ be a discrete group of transformations acting on the hyperbolic space $ B ^ {n} $ of dimension $ n $. A family of probability measures $ \{ {\mu _ {x} } : {x \in B ^ {n} } \} $( cf. Probability measure) supported on the limit set $ L ( \Gamma ) $ is called $\alpha$-conformal if for every $ \gamma \in \Gamma $ and every $ x,y \in B ^ {n} $ the measures $ \mu _ {x} $ and $ \mu _ {y} $ are equivalent with Radon–Nikodým derivative (cf. also Radon–Nikodým theorem)

$$ { \frac{d \mu _ {x} }{d \mu _ {y} } } ( \xi ) = \left ( { \frac{P ( x, \xi ) }{P ( y, \xi ) } } \right ) ^ \alpha , $$

where $ P ( z, \xi ) = { {( 1 - \| z \| ^ {2} ) } / {\| {z - \xi } \| ^ {2} } } $ denotes the Poisson kernel (cf. Poisson integral).

Conformality of the measure can be described by restricting to $ y = \gamma ( x ) $, where $ \gamma $ is any Möbius function leaving $ B ^ {n} $ invariant. In this case the definition reads:

$$ \mu _ {\gamma ^ {- 1 } ( x ) } ( \gamma ( E ) ) = \int\limits _ { E } {\left | {\gamma _ {x} ^ \prime ( \xi ) } \right | ^ \alpha } {\mu _ {x} ( d \xi ) } , $$

where $ \gamma _ {x} ^ \prime ( \xi ) = {P ( \gamma ^ {- 1 } ( x ) , \xi ) } / {P ( x, \xi ) } $ and $ E $ is any measurable set. If $ \gamma \in \Gamma $ and $ \mu _ {\gamma ^ {- 1 } ( x ) } = \gamma ^ {*} \mu _ {x} $, then $ \mu = \mu _ {x} $ is a conformal measure in the following sense (see [a1]).

Let $ T $ be a measurable transformation (cf. Measurable mapping) acting on the measure space $ X $ with $ \sigma $- algebra $ {\mathcal B} $. A measure $ \mu $ is called conformal for the function $ \varphi : X \rightarrow {\mathbf R _ {+} } $ if for every set $ E \in {\mathcal B} $ on which $ T $ acts as a measurable isomorphism,

$$ \mu ( T ( E ) ) = \int\limits _ { E } {\varphi ( \xi ) } {\mu ( d \xi ) } . $$

Thus, $ m $ is conformal for $ \varphi : X \rightarrow {\mathbf R _ {+} } $ if and only if the Jacobian of $ m $ under $ T $ is given by $ \varphi $. The Frobenius–Perron operator $ P $ can be defined as the restriction to $ L _ {1} ( m ) $ of the dual operator on $ L _ \infty ^ {*} ( m ) $, $ \int {Pf \cdot g } {dm } = \int {f \cdot g \circ T } {dm } $, and it satisfies $ P ^ {*} m = m $. In many examples $ P $ can be written explicitly in the form

$$ Pf ( x ) = \sum _ {T ( y ) = x } f ( y ) \varphi ( y ) ^ {- 1 } , $$

and this representation permits the application of Ruelle's thermodynamic formalism [a3]. The importance of the notion of a conformal measure can be seen from this.

Besides its use in the ergodic theory of dynamical systems and statistical mechanics (including discrete groups and geodesic flows), it allows one to study geometric and number-theoretic problems like fractal dimensions, Diophantine approximations and recurrence.

References

[a1]	M. Denker, M. Urbański, "On the existence of conformal measures" Trans. Amer. Math. Soc. , 328 (1991) pp. 563–587
[a2]	S.J. Patterson, "The limit set of a Fuchsian group" Acta Math. , 136 (1976) pp. 241–273
[a3]	D. Sullivan, "The density at infinity of a discrete group of hyperbolic motions" IHES Publ. Math. , 50 (1979) pp. 171–202
[a4]	D. Ruelle, "Thermodynamic formalism" , Encycl. Math. Appl. , 5 , Addison-Wesley (1976)