Abstract:

The main goal is to explain the paper entitled "The (3n+1)-problem and holomorphic dynamics" (S. Letherman, D. Schleicher and R. Wood, 1998) in which the authors gave a possible translation of the main conjecture about the (3n+1)-problem in the language of complex dynamics. The (3n+1)-map acting on $mathbb N$ is the following: If $n$ is even the image is $n/2$, if $n$ is odd the image is $3n+1$. The conjecture is that for all $n_0inmathbb N$, after a finite number of iterates the process reaches the number $n=1$. The idea is to construct a (family of) transcendental entire function(s) which coincide with the (3n+1)-map (defined in $mathbb Z$) and then re-stated the (3n+1)-conjecture in terms of existence (or not) of certain Fatou components.