[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
pathological:
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.
Jacques
http://www.cas.mcmaster.ca/~carette
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