View source for Conformal measure

The original definition of a conformal measure (or density) is due to S.J. Patterson [[#References|[a2]]] in the case of a [[Fuchsian group|Fuchsian group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103501.png" /> acting on the hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103502.png" /> (cf. [[Poincaré model|Poincaré model]]). Its definition carries over to any dimension (D. Sullivan, [[#References|[a4]]]):

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103503.png" /> be a [[Discrete group of transformations|discrete group of transformations]] acting on the hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103504.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103505.png" />. A family of probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103506.png" /> (cf. [[Probability measure|Probability measure]]) supported on the limit set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103507.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c1103509.png" />-conformal if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035010.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035011.png" /> the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035013.png" /> are equivalent with Radon–Nikodým derivative (cf. also [[Radon–Nikodým theorem|Radon–Nikodým theorem]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035015.png" /> denotes the Poisson kernel (cf. [[Poisson integral|Poisson integral]]).

Conformality of the measure can be described by restricting to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035017.png" /> is any [[Möbius function|Möbius function]] leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035018.png" /> invariant. In this case the definition reads:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035021.png" /> is any measurable set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035024.png" /> is a conformal measure in the following sense (see [[#References|[a1]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035025.png" /> be a measurable transformation (cf. [[Measurable mapping|Measurable mapping]]) acting on the [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035027.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035028.png" />. A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035029.png" /> is called conformal for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035030.png" /> if for every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035031.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035032.png" /> acts as a measurable isomorphism,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035033.png" /></td> </tr></table>

Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035034.png" /> is conformal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035035.png" /> if and only if the Jacobian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035036.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035037.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035038.png" />. The Frobenius–Perron operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035039.png" /> can be defined as the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035040.png" /> of the dual operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035042.png" />, and it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035043.png" />. In many examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035044.png" /> can be written explicitly in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110350/c11035045.png" /></td> </tr></table>

and this representation permits the application of Ruelle's thermodynamic formalism [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Denker,   M. Urbański,   "On the existence of conformal measures"  ''Trans. Amer. Math. Soc.'' , '''328'''  (1991)  pp. 563–587</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.J. Patterson,   "The limit set of a Fuchsian group"  ''Acta Math.'' , '''136'''  (1976)  pp. 241–273</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sullivan,   "The density at infinity of a discrete group of hyperbolic motions"  ''IHES Publ. Math.'' , '''50'''  (1979)  pp. 171–202</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Ruelle,   "Thermodynamic formalism" , ''Encycl. Math. Appl.'' , '''5''' , Addison-Wesley  (1976)</TD></TR></table>