The term sawtooth snap has been coined in connection with the replacement theory of time.  It describes any temporal anomaly in which more than two histories cause each other in sequence, such that each causes the next.

Image © E. R. Jones and M. Joseph Young.  Used courtesy Valdron Inc

For a simple case of a sawtooth snap, imagine that on Friday morning an experimenter checks his strongbox and finds it empty, then that night uses a time machine to send one hundred dollars back in time to appear in that box that morning.  In the revised history, on Friday morning he checks the box, finds and removes the money, spends one dollar on a soda, and that night sends the rest back to Friday morning.  On each successive replay of that Friday the experimenter removes the money from the box, spends a dollar, and sends the remainder back to that morning.  Thus each morning after the first the experimenter will find less money in the box, because each evening he sends less back.  The history of the world changes incrementally with each iteration, by a decrease of one dollar in the box.

Sawtooth snaps technically have three distinct forms, distinguished by their terminations.  In the example above, eventually there will be a single dollar in the box, which the experimenter will spend on soda, and then he will send no money to the future; his counterpart that morning will find an empty box and will assume that the experiment has not yet begun, so that night he will send one hundred dollars to the past, restarting the sequence.  This is called an infinity loop termination, because it is like an infinity loop but that it involves multiple timelines in its progression.  It is possible for a sawtooth snap to resolve to an N-jump (to be discussed next time), a self-supporting single history in which the last timeline causes itself and time continues based on that final history.  It is also theoretically possible for a sawtooth snap to continue in a non-repeating causal chain, such that the end of each timeline initiates a unique new timeline (akin to the decimal string on an irrational number).

The term Cycling Causality has also been used to describe any anomaly that resembles a sawtooth snap, but particularly those which have infinity loop terminations, in which a string of several ("2 to n") distinct histories cause each other in a repeating sequence.  It is thus commonly connected to sawtooth snaps, but is a distinct term in that it includes simple infinity loops but excludes perpetual sawtooth snaps (which never repeat a duplicate history) and those which resolve to an N-jump termination.  The use of this term is not consistent, though, sometimes being used to describe any sawtooth snap.

Many movies are best explained as sawtooth snaps, including the first two Terminator films.

Next time we will discuss the N-jump.