Staircase baker's map generates flaring-type time series

The baker's map, invented by Eberhard Hopf in 1937, is an intuitively acces ible, two-dimensional chaos-generating discrete dynamical system. This map, which describes the transformation of an idealized two-dimensional dough b y stretching, cutting and piling, is non-dissipative. Nevertheless the "x" variable is identical with the dissipative, one-dimensional Bernoulli-shift -generating map. The generalization proposed here takes up ideas of Yaacov Sinai in a modified form. It has a staircase-like shape, with every next st ep half as high as the preceding one. Each pair of neighboring elements exc hanges an equal volume (area) during every iteration step in a scaled manne r. Since the density of iterated points is constant, the thin tail (to the right, say) is visited only exponentially rarely. This observation already explains the map's main qualitative behavior: The "x" variable shows "flare s". The time series of this variable is closely analogous to that of a flar ing-type dissipative dynamical system - like those recently described in an abstract economic model. An initial point starting its journey in the tale (or "antenna", if we tilt the map upwards by 90 degrees) is predictably at tracted by the broad left hand (bottom) part, in order to only very rarely venture out again to the tip. Yet whenever it does so, it thereby creates, with the top of a flare, a new "far-from-equilibrium" initial condition, in this reversible system. The system therefore qualifies as a discrete analo gue to a far-from-equilibrium multiparticle Hlamiltonian system. The height of the flare hereby corresponds to the momentary height of the H function of a gas. An observable which is even more closely related to the momentary negative entropy was recently described. Dependent on the numerical accura cy chosen, "Poincare cycles" of two different types (periodic and nonperiod ic) can be observed for the first time.