Thermodynamic Dictionary for Syracuse-like Maps

In this paper we estabilsh a dictionary between the behavior of a general map $f: X \to X$ which is defined by using a decomposition into disjoint subsets $X = A \cup B$ and the thermodynamic formalism. The approach of this paper is set theoretical, topological and ergodic. We define a key topology and consider its Borel $σ$-algebra in order to study the thermodynamic formalism for the general setting, proving that recurrence implies periodicity, the topological entropy is zero, we can decompose the invariant measures into a general sum of ergodic ones and we establish a dictionary between the cycles and the equilibrium states of continuous potentials. All this is done for the general setting, with applications for the Syracuse maps and the Collatz map in particular, as a significant application of the dictionary, proving the finiteness of cycles. Finally, we prove that any Syracuse map has absence of divergent orbits, proving the weak Collatz Conjecture.