Furstenberg Topology and Collatz Problem

The aims of this paper are two-fold. First, we present the result of the decomposition on the iterations of a Collatz transform into arithmetic sequences. With this, we prove that in Furstenberg topology, the set of (odd) integers with an infinite stopping time is closed and nowhere dense. Then, we move our considerations to some monoids L in N, where we define a suitably modified Collatz transform, and we present some results of numerical investigations on the behaviour of these modified transforms.