[FOM] Category and Measure

[Lasse Rempe]

Lasse Rempe wrote:
> 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))...)).
>   
On second thought, the following is perhaps an even more striking 
example. Take

f(z) = pi*sinh(z) = pi/2 * (exp(z) - exp(-z)).

This map also has the Julia set being the whole plane (this is easy to 
see), and hence all points except for a meager set have dense orbits. On 
the other hand, *almost every* point z tends to infinity under iteration 
of f. (This follows from a result of McMullen, together with standard 
methods.)

So the limiting behavior of "almost every point" and "all but a meager 
set of points" are as different as one could possibly imagine.

Lasse

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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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