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 acting on the hyperbolic space
(cf. Poincaré model). Its definition carries over to any dimension (D. Sullivan, [a4]):
Let be a discrete group of transformations acting on the hyperbolic space
of dimension
. A family of probability measures
(cf. Probability measure) supported on the limit set
is called
-conformal if for every
and every
the measures
and
are equivalent with Radon–Nikodým derivative (cf. also Radon–Nikodým theorem)
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where denotes the Poisson kernel (cf. Poisson integral).
Conformality of the measure can be described by restricting to , where
is any Möbius function leaving
invariant. In this case the definition reads:
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where and
is any measurable set. If
and
, then
is a conformal measure in the following sense (see [a1]).
Let be a measurable transformation (cf. Measurable mapping) acting on the measure space
with
-algebra
. A measure
is called conformal for the function
if for every set
on which
acts as a measurable isomorphism,
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Thus, is conformal for
if and only if the Jacobian of
under
is given by
. The Frobenius–Perron operator
can be defined as the restriction to
of the dual operator on
,
, and it satisfies
. In many examples
can be written explicitly in the form
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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) |
Conformal measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_measure&oldid=46457