Syracuse Maps as Non-Singular Power-Bounded Transformations and Their Inverse Maps

The paper’s Theorem 3 claims: for any fixed finite measure µ equivalent to counting measure on N, power-boundedness in L1(µ) is equivalent to “T has a cycle and every orbit hits some cycle.” The forward implication (power-bounded ⇒ eventual periodicity) is correct and their proof is essentially sound (see their argument selecting N with µ(T^N x) small to contradict power-boundedness if the forward orbit is infinite) . However, the converse as stated is false. The paper’s proof constructs a new measure tailored to the inverse trees of the cycles and shows power-boundedness for that constructed measure, not for an arbitrary given µ; this establishes an existence-of-measure statement (as in their Theorem 2) rather than the for-all-µ claim in Theorem 3 and the abstract . The candidate solution provides a concrete counterexample: a map where every orbit reaches a fixed point in one step, yet there exists a finite µ equivalent to counting making sup_A µ(T^{-1}A)/µ(A) unbounded, so power-boundedness fails. This invalidates the theorem’s ⇐ direction for arbitrary µ, and aligns with the correct, µ-dependent criterion sketched by the model.