Abstract:
The Ulam-Kakutani Collatz Conjecture
The Ulam-Kakutani Collatz conjecture explores the behavior of sequences generated
by the Collatz function, defined as follows: For any positive integer n,
f(n) =
8><>:
n2
; if n is even;
3n + 1; if n is odd:
A topological approach to this conjecture involves analyzing the set of integers
as a discrete space and examining the dynamics of the function through its iterative mappings. The key aspects of this approach include: Mapping and fixed points,
Topological dynamics, Compactness, Connectedness, and Orbit structure. This approach aims to provide insights into the underlying structure of the conjecture and
its potential resolution through topological methods.
In addition to this, the researcher proposes to prove that if (N; τf) is not a w-Ro
space, it is equivalent to saying that the conjecture is true. Here, τf is the topology
on N given by the open sets as those subsets θ of N such that f−1(θ) ⊂ θ, where f
is the Collatz function.
