On the Collatz Conjecture: Topological and Ergodic Approach

We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel sigma-algebra we show that recurrence implies periodicity. Moreover, we establish that the set of periodic orbits is finite if, and only if, every continuous potential possesses some equilibrium state. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. Additionally, by using the dictionary established in the paper, we prove finiteness of cycles, which is a significant advance to the conjecture itself. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result on the finiteness of orbits for a general class of important maps.