Animation of Generalized Baker Map

Juraj Kumičák

The applets below are meant for exploration of the properties of Generalized Baker Map (GBM). What you see inside them, is the unit square E with points upon which the GBM is acting iteratively. The animation starts in the state n = 0 and will run (after pressing Run button) by default in the direction of growing iteration index n (the numbers n are seen inside the square in the left upper corner).

In both applets you can choose the parameter w (acceptable values are 2 £ w £ 9). The latter defines the local contraction of the GBM. Setting w = 2 you will obtain locally measure preserving map, referred to as classical Baker Map.

After changing any values, please, Reset the applet. To run the iteration, click on Run (you can stop it by clicking on Step/Stop). The iterations will change by default approximately after every second. Clicking on Speed (this is a toggle switch), you will get the highest possible speed, limited by the power of your computer. Clicking repeatedly on Step you can watch single frames.

Remarks: The animation is not foolproof yet. Coordinates of single points have to be input as fractions: use e. g. 0/10 insted of 0. During long iterations with large number of points, false isolated points may exceptionally appear. On rare occassions the background can change color (to blue) when going through n = 0. In the latter case, pressing the Step/Stop button will always help.

You can experiment with two applets:

Evolution of random sets of points
Evolution of single points

Evolution of random sets of points

The initial state is chosen here as a set of points randomly filling a rectangle. You can choose the dimensions of the rectangle by setting its horizontal (H) and vertical (V) boundaries in the interval [0..100] (the percentage of the side of the unit square). You can also choose the number of points in the rectangle. (The default subset generates Fig. 11 in the paper).

The grid is added to enable approximate localization of points in E. The grid spacing is set to 1/w in order to visualize the structural similarity of sets approaching attractors (and departing repellors) for different values of the parameter w.

The most important is the Reverse switch which you can use any time. It will cause running the iteration backwards and you should convince yourself that you will come to the original state after arbitrary number of iterations.

The unit square E could be considered as representing the configuration space of one-dimensional gas. The red and green points represent projections of the points onto diagonals. The paper explains the reasons why we can consider red points to represent positions of the "molecules" of the 1D gas (and eventually the green ones as a kind of their "velocities").

The blue points behave strictly reversibly and the behavior of the red ones can give the observer an idea about the origin of irreversibility in such strictly reversible model system. By viewing the animation one should arrive at an understanding that irreversibility appears as the result of discarding the information about "velocities". Note also, that positive values of n have plausible meaning of future states, whereas the negative ones can be considered as past states, this giving evident explanation of the notion of absolute age, introduced by the Brussels school.

The behavior, violating the second law of thermodynamics can be observed in the following way. Reverse the initial set and iterate it for, say, n = −20 steps. Then press the Reverse switch again and observe the forward evolution. For n steps it will violate the law, at the step n = 0 the evolution obeying the law will start.

N. B. Reversibility requires to avoid rounding errors, which in its turn necessitates the use of rational arithmetics, i.e. to calculate with integers and their fractions. The iterated points are expressed after n iterations by numerators and denominators of the order w ⁿ. After thousands of iterations these are huge numbers. To avoid overflow I recommend to limit the number of iterations to less than 10 000 and number of points to less than 5000.

Evolution of single points

The initial state is chosen here as a single point with rational coordinates. The newly calculated point is positioned into the center of a red square, the previous one into the center of the green square – this should enable their tracking. The animation keeps iterated points on the screen but erases them when running over them repeatedly upon reversal. The initial point is framed blue.

Opening the Java Console you may watch the calculated "absolute age" τ – see Eq. (20) in the paper – together with the rational coordinates of the iterated point. Notice how the fractions expressing coordinates are growing when departing from the present – both in the direction of past and future. You can also see how closely τ follows the iteration index k.

You can try to iterate arbitrary point up to, say, n > 20, then reverse the iteration until n < −20, and upon the reversal you should be able to see the departure from repellor in the past (negative n) and approach to the attractor in the future (positive n). Choosing an initial point with y = 0 you will obtain points lying exactly on the attractor. (The default point generates the Fig. 7 in the paper).

The animation enables to watch the approach to periodic cycles. The classical map, corresponding to w = 2, is rich in such cycles, since any point with odd denominator of x-coordinate will approach some cycle for arbitrary y-coordinate. Similarly, any point with odd denominator of y-coordinate will depart from some cycle. For w > 2, however, we do not have such simple rule. Nevertheless, for any w there exist periods of any length. Some examples are given in the following table. By the "structure" is meant the sequence in which the points of the cycle visit left and right side of the dividing line. The coordinates expressed by integers exceeding 5 numerals are denoted by *.

x-coordinates of some points which approach cycles

Period	w = 2	w = 3	w = 4	w = 5	Structure
2	1/3	4/7	9/13	16/21	LR
3	2/7	12/23	36/55	80/109	L2 R
	5/7	22/25	57/61	116/121	L R2
4	1/15	16/73	81/129	256/561	L3 R
	7/15	52/79	189/253	496/621	L R3
	1/5	32/77	135/247	128/203	L2 R2
5	1/31	32/227	243/943	1024/2869	L4 R
	7/31	104/239	81/145	1984/3109	L2 R3
	9/31	124/235	657/997	2256/3061	L R L2 R
	11/31	140/239	711/1015	2384/3109	L R L R2
6	1/9	208/721	1701/4069	7936/15561	L3 R3
	5/63	176/713	1539/4015	7424/15369	(L R)2 L2
	11/63	401/103	2133/4069	9536/15561	L R2 L R L
7	5/127	352/2155	4617/16141	29696/77101	(L R)2 L3
	11/127	560/2171	6399/16303	38144/77869	L R L3 R2
	13/127	608//2171	6723/16303	38144/77869	L R2 L3 R
	43/127	1252/2179	11349/16357	59536/78061	(L R)2 L R2
8	19/255	1552/6529	24381/65293	*	L3 R2 L2 R
	29/255	16/55	27459/65455	*	L3 R L R3
	53/255	2768/6545	36099/65455	*	(L R)3 R L
	91/255	3844/6553	45909/65509	*	(L R2)2 L R
9	11/73	7048/19651	*	*	(L2 R)2 L R2
	13/73	7688/19667	*	*	L R L2 R2 L R2
	55/511	5600/19667	*	*	L3 R3 L R2
	85/511	7480/19651	*	*	(L R)4 L
	171/511	11252/19667	*	*	(L R)4 R
	183/511	11548/19675	*	*	L R (R 2 L)2 R
10	71/1023	13328/58985	*	*	L3 R3 L3 R
	343/1023	33788/59033	*	*	(L R)4 R2

Combining an x-coordinate x₁ approaching a p₁-period with x-coordinate x₂ approaching a p₂-period into the new coordinate (x₁, 1 – x₂), one obtains trajectory unwinding from a p₂-cycle in the past and approaching a p₁-cycle in the future. To watch such trajectories, I give a short list of points moving from past limit cycle to the future limit cycle.

Selected points generating pairs of limit cycles

w	X	Y	Past cycle	Future cycle
2	6/7	30/31	5	3
	1/18	1/10	4	6
	64/125	42/54	6	100
	256/401	60/73	9	200
	502/2189	106/127	7	990
3	18/23	39/103	6	3
	16/55	45/77	4	8
	16/73	63/103	6	4
	16/73	45/239	4	5
	32/77	63/103	6	4
	96/235	27/79	4	5
	7480/19651	39/55	8	9
4	9/13	11524/16141	2	7
	81/229	112/247	4	4
5	116/121	725/3109	5	3

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