[FOM] Category and Measure
joeshipman at aol.com
Tue Sep 18 11:26:10 EDT 2007
From: Lasse Rempe <L.Rempe at liverpool.ac.uk>
Famous examples come up in the "center problem". E.g., consider a
polynomial of the form
f(z) = lambda*z + a_2 z^2 + ... + a_n z^n
with lambda = exp(i*theta) on the unit circle. The question is whether
the fixed point 0 is a "center"; i.e., whether f, near 0, is conjugate
to the rotation
R(z) = lambda*z.
The answer is "yes" for almost every rotation number theta (due to
Siegel), but "no" for all but a meager set of thetas (due to Cremer).
Here is another example. Consider the function f(z) = exp(z). We study
the behavior of the iterates f^n(z) = f(f(...(f(z))...)).
A famous result of Misiurewicz (conjectured by Fatou) says that the
Julia set of f is the entire plane, which means, in particular, that
all but a meager set of points z the sequence f^n(z) is dense in the
plane. However, by a result of Lyubich and Rees, almost every orbit has
a very different behavior: it accumulates exactly on the sequence
0,1,e,e^e,e^e^e, ... .
These are both excellent examples -- they show that the question of
whether a "typical" theta gives a center, or whether a "typical" orbit
is dense, is essentially unanswerable, even when we completely
understand the mathematical situation.
There is something very mysterious going on here, which leads me to
suspect that one of the other of the two notions of "typical" (co-null
or co-meager) is ultimately more fundamental and applicable in "real
life" than the other. For my money, measure seems a more robust concept
than category as far as applications to physics are concerned, but I'd
be happy to hear an opposing view.
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