[FOM] Category and Measure

Lasse Rempe L.Rempe at liverpool.ac.uk
Tue Sep 18 05:58:43 EDT 2007

joeshipman at aol.com wrote:
> But, can someone explain what's so useful about meager sets when 
> working with a measure space like the real numbers?

Meager sets and nullsets both routinely come up in the theory of 
dynamical systems, and often play complementary roles. Generally, you 
think of meagre sets of being "topologically small", while nullsets are 
"measurably small".

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 for 
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 may not be exactly elementary examples, but perhaps they 
illustrate that both concepts have their uses, but are rather orthogonal 
to each other.

Dr. Lasse Rempe
Dept. of Math. Sciences, Univ. of Liverpool, Liverpool L69 7ZL
Office 505; tel. +44 (0)151 794 4058, fax +44 (0)151 794 4061 

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