# [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.

--
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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
http://pcwww.liv.ac.uk/~lrempe
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