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 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 if every continuous potential with finite pressure possesses some equilibrium state then we have either finiteness of cycles or infinitely many cycles sharing the same period. The existence of some continuous potential with no equilibrium state is equivalent to the unboundedness of periods of cycles. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. We also prove that we have either infinitely many cycles or no divergent orbits. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result for this general class of important maps.