[FOM] constructivism and physics

Jacques Carette carette at mcmaster.ca
Tue Feb 7 21:28:24 EST 2006

John McCarthy wrote:
> I wouldn't be surprised if chaos theory got itself into non-measurable 
> sets.
I can't claim "non-measurable", but in my PhD thesis I give an example 
of a natural metric extracted from a dynamical system which is fairly 
1) it is in L^1(S), with S = Riemann Sphere
2) it defines a topology for S equivalent to the one given by the usual 
spherical metric
3) its density function has a singularity at a dense set of points in S
4) for that metric, the underlying dynamical system (a rational 
function) is *expanding*
5) the Julia set of that dynamical system is the whole of S.

The post-critical orbit of this rational function is dense in S, and 
thus any metric which makes this expanding necessarily has a singularity 
at each point of the post-critical orbit.  That such a metric exists at 
all was suprising at the time.



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