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In this paper free harmonic analysis tools are used to study paraboliciteration in the complex upper half-plane. The main result here is a completecharacterization for the norming constants in the monotonic central limittheorem. This allows us to construct a new class of conservative and ergodicmeasure-preserving transformations on the real line with Lebesgue measure.Among all, we mention that the generalized Boole transformation with infinitelymany poles is shown to be conservative, as long as its residues are summable.

The central limit theorem for monotone convolution with applications to free Levy processes and infinite ergodic theory