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Fixed point combinators (and their generalization: looping combinators) areclassic notions belonging to the heart of lambda-calculus and logic. We startwith an exploration of the structure of fixed point combinators (fpc's), vastlygeneralizing the well-known fact that if Y is an fpc, Y(SI) is again an fpc,generating the Boehm sequence of fpc's. Using the infinitary lambda-calculus wedevise infinitely many other generation schemes for fpc's. In this way we findschemes and building blocks to construct new fpc's in a modular way.  Having created a plethora of new fixed point combinators, the task is toprove that they are indeed new. That is, we have to prove theirbeta-inconvertibility. Known techniques via Boehm Trees do not apply, becauseall fpc's have the same Boehm Tree (BT). Therefore, we employ `clocked BT's',with annotations that convey information of the tempo in which the data in theBT are produced. BT's are thus enriched with an intrinsic clock behaviour,leading to a refined discrimination method for lambda-terms. The correspondingequality is strictly intermediate between beta-convertibility and BT-equality,the equality in the classical models of lambda-calculus. An analogous approachpertains to Levy-Longo Berarducci trees. Finally, we increase thediscrimination power by a precision of the clock notion that we call `atomicclock'.

Modular Construction of Fixed Point Combinators and Clocked Boehm Trees