Normally one would expect that time evolution of a physical system
is described by a differential equation. That is how laws of
physics are usually expressed. Here however we have a surprise: in
EEQT a history of an * individual quantum system*, coupled to a
classical system, as in every real world * experiment*, is
described by a * process* or an * algorithm* rather than
by a differential equation. The PDP is similar to those studied in
the science of economics, where periods of smooth fluctuations are
interrupted by market crashes
[13,14]. A quantum jump is what corresponds to a
market crash - a discontinuity, a 'catastrophe'. But
discontinuities and catastrophes have their own laws, and here
comes the concept of a * piecewise deterministic Markov
process*^{} - a PDP.

PDPs are the simple and elegant ways to describe the world in
terms of cyclic, rather than linear, time; that is the world of
cyclically, though somewhat irregularly, recurring catastrophes.
In the cases studied in the present paper the catastrophes come
from the coupling of a quantum system to a system of two-state
"detectors". In this case the catastrophes are not really
catastrophes for the detectors - detectors just flip, which is
exactly what detectors do for living. But these flips bring
catastrophes for the quantum system, because with each flip of the
detector, with each "event", as we call it, the quantum system
state vector breaks its continuous evolution, and instantaneously
jumps to a different state - it "rejuvenates" and it starts
another cycle of a peaceful, continuous evolution - till the next
catastrophe.^{}

The EEQT algorithm generating quantum jumps is similar in its
nature to a nonlinear iterated function system (IFS)
[15] (see also [16] and references therein)
and, as such, it generically produces a chaotic dynamics for the
coupled system. Here the probabilities assigned to the maps are
derived from quantum transition probabilities and thus depend on
the actual point, but such generalizations of the IFS's have been
also studied (cf.
[17] and references therein). In the present paper we
describe the algorithm generating * quantum
fractals*^{}, that is self-similar patterns on the projective plane
, when a continuous in time
"measurement" of several spin directions at once takes
place.^{} As stated above the algorithm of EEQT describes a
piecewise deterministic random process - periods of a smooth
evolution interspersed with catastrophic "jumps." Of course, once
we have individual description, we can also get the laws for
statistical ensembles. Here we get a nice, linear, Liouville
evolution equation for measures - as it is usual in studying
chaotic dynamics. The fact that there is a unique PDP generating
the Liouville equation in the framework of EEQT has been proven in
Ref. [21].