1. Introduction

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].