FIRST-RETURN MAPS IN A SYSTEM OF TWO ROTATIONS AREK GOETZ
One basic goal of the mathematical theory of dynamical systems is to determine or characterize the long-term behavior of the system. For example, a Euclidean rotation T in the plane has simple dynamics. For every point x other than the center of rotation T the orbit fx; Tx; T (Tx); : : : g is a nite set or it is a dense subset of a circle. This depends on whether T is a rotation by an angle commensurate with the full angle. Composing two rotations in the plane leads to complicated and fascinating dynamical systems. These systems are also surprising, for they can have attractors, repellers, and self-similar structures which are usually observed in systems whose generating map has either contacting or expanding properties. Let us partition the plane into two half-planes P0 = f(x; y) : x < 0g and P1 = f(x; y) : x 0g, and x two arbitrary rotations T0 and T1, each with a di erent center of rotation. De ne T : R 2 ! R2 to be: Tx = TT0xx ifif xx 22 PP0:; 1 1 Even though the de nition of T is very elementary, the dynamical systems that T generates can be very complicated, and in general, is is not well understood. In this article, we illustrate how the studies of the rst-return map give insight into a few particular cases of the dynamical system T : R 2 ! R 2 . We brie y describe three examples: an attracting system, a repelling system, and a new attracting system with a self-similar structure of periodic domains on its attractor which resembles Sierpinski gasket. We then invite the reader to conjecture and prove other theorems about this special class of piecewise isometries. Computer modeling may serve as an illustration, veri cation and experimental tool. It is available to the reader in the form of a java applet which is located at http://math.bu.edu/people/goetz/Research/Att.html. Computer graphics shows that the dynamics of piecewise rotations is intriguing and very beautiful (Figure 2(a) and 4). However, more importantly, these systems appear in other contexts as well. Systems of piecewise rotations have been recently connected to the dynamics of electronic components called digital lters [1, 14]. The theory of digital lters are of particular interest to the electrical engineering community as ltering of frequences using microprocessors became inexpensive and it is now widely used in the industry. The systems of piecewise rotations also appear as return maps in dual biliards [16]. They are generalizations of interval exchange maps [12], interval translations [2], partially de ned rotations of the circle [13] or polygon exchanges [9, 10]. Finally, the example studied in this article are very special cases of piecewise isometric systems [5, 6]. Date : March 9, 1999. .
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Example of an attracting system. A number of single points denote the orbit of the point x. The small digits denote successive iterates of x. Each lattice mesh follows the same pattern of visits to P0 and P1, and it is eventually trapped in one of the two centered shaded squares.
Figure 1.
Example 1. Attracting system. In our rst example, the centers of rotation of T0 and T1 lie on the X ?axis (Figure 1 and 2(a)). In this example, all orbits are monotonically attracted to the set A of all points whose iterates are trapped in one of the sets P0 or P1. We start with a brief analysis of a very basic system illustrated in Figure 1. Proposition 1. Let T0 be the rotation about S0 = (?1; 0) by 0 = 90 and let T1 be the rotation about S1 by 1 = 90 . Then all orbits under T are nite and they are trapped in union of the two squares with side 2, centered at S0 and at S1 . Proof. Let jjajj denote the distance between a and the closest integer. Observe that T0 (x0 ; y0) = (?y0 + 1; x0 + 1) and T1 (x0 ; y0) = (?y0 ? 1; x0 ? 1), the pair fjjx0jj; jjy0jjg remains unchanged under the application of the piecewise rotation T to (x0 ; y0). It follows that every orbit under T is discrete. Also, note that since the centers of rotation S0 and S1 are symmetric with respect to the Y -axis, the distance between an iterate and the set fS0 ; S1g remains constant or it decreases under the application of T . This implies that all orbits are bounded. Bounded and discrete orbits, must be nite, hence all points are eventually periodic (a point x is eventually periodic if for some m and n, T m (T nx) = T nx). This means that the distance between periodic iterates and fS0; S1g remains constant under the application of T . Observe that an orbit may not alteranate between the half-planes. For otherwise, this orbit must visit Y P1. However, then the distance between the iterate visiting Y and fS0; S1g decreases under the application of T . Hence every oribt must be trapped in P0 or in P1 . More precisely, all orbits are eventually attracted to the union of the two squares with side 2, centered at S0 and at S1 . It turns out that some of the above ideas can be used in analyzing more general cases. Let Ti (i = 0; 1) be the rotation by angle i about Si. Suppose that S0 2 Int(P0) and S1 2 Int(P1),
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(a) Mosaic of cells of a symmetric attracting system (maximal regions following the same pattern of visits to P0 and P1). (b) Single attracting orbit. The iterates visiting P0 get closer and closer to S0. This orbit is eventually trapped in the disc D1 . The angles of rotation 0 and 1 are non-commensurate with the full angle. Figure 2.
that is both points S0 and S1 are xed under T . Also, xed under T are the two discs D0 and D1 tangent to the Y ?axis and centered at S0 and S1 , respectively. First, let us note again that there are two possibilities for the behavior of an orbit. Either an orbit of a point is eventually trapped in one of the half-planes P0 , or P1, or it visits both half-planes in nitely often. The dynamics of orbits which are eventually trapped in one of the half-planes is especially simple as it is equivalent to the dynamics of the rotation on a circle. Suppose that there exists some point x 2 P0 whose orbit alternates between P0 and P1. Also, suppose for the moment that S0 is symmetric to S1 with respect to Y . Let x1 = T m x be the rst iterate of x which \enters" P1. Then dist(x; S0 ) dist(x1 ; S1 ). Applying this inequality twice, we obtain (1) dist(x; S0) dist(TP0 x; S0 ); where TP0 denotes the rst-return map to P0. In other words, the orbit of x gets closer and closer to the center of rotation S0. Figure 2(a) suggests that inequality (1) also holds in the more general case when S0 2 P0 and S1 2 P1 are no longer symmetric (I nd it surprising). This, in fact, is true and its veri cation is an analytic calculation [7]. Using observation (1), we then argue that the iterates of x visiting P0 must accumulate outside D0 , on some circle C0 centered at S0. A more detailed analysis [7] yields that this \limit" circle C0 cannot intersect Y in two points, hence C0 must be the boundary of the disc D0. This implies that the iterates of x visiting the right half-plane P1 must accumulate on the boundary of the disc D1. (Figure 2(a)). Our analysis maybe potentially vacuous for we have assumed the existence of an orbit visiting both P0 and P1 in nitely often. In fact, orbits visiting P0 and P1 in nitely often exist, however, this is the case only when 0 and 1 are both non-commensurate with the full angle. Otherwise, all orbits are eventually trapped in one of the half-planes [7].
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The dynamics of T can be brie y summarized as follows. Let A be the set of all points whose orbits are trapped in one of the half-planes P0, or P1 . (The set A is either the union of two discs, the union of a disc and a regular polygon, or it it the union of two regular polygons.) Proposition 2. Let T be a piecewise rotation with centers of rotation: Si 2 Int(Pi) (i = 0; 1) and angles of rotation i. Suppose that the line S0 S1 is perpendicular to the discontinuity line Y . Then, for all x 2 R 2 , dist(T nx; A) ! 0 as n ! 1; regardless of the choice of 0 and 1. The system T : R 2 ! R 2 divides the plane into regions following the same pattern of visits to P0 and P1 (Figure 2(a)). These regions correspond to the symbolic squences obtained by recording the indices of the half-planes visited by an orbit. For instance, since every point x 2 D1 is trapped in P1, the sequence associated to x is the in nite sequence of ones. On the other hand, if the angle 1 is non-commensurate with full angle, then the set h11 : : : i of all points with codings consisting of ones is the disc D1. Also, an orbit which is not trapped in one of the atoms gives rise to a sequence which consists of alternating blocks of zeroes and ones 0 : : : 01 : : : 10 : : : 01 : : : . Coding of orbits is a very powerful tool in the study of the behavior of orbits [3, 11, 15]. Formally, the coding map is : R 2 ! f0; 1gN such that (x) = w0w1 : : : , where T k x 2 Pwk and wk 2 f0; 1g. The sets of points following the same coding will be called cells. In this article, every cell is a convex set for it is an intersection of half-planes [7]. We say that a cell is periodic if its coding is a periodic sequence. It is called eventually periodic if its coding consists of a nite block of digits (possibly of length zero) followed by a periodic sequence. We invite the reader to show, for example, that for the choice of the parameters from Proposition 2, the mosaic of cells coincides with the square lattice illustrated in Figure 1. Our next example illustrates that the parameters de ning T can be chosen so that all orbits which are not trapped in one of the half-planes P0 or P1 , must diverge to in nity (Figure 3(a)). Example 2. Repelling system. Let S0 = (?1; 0), S1 = (1; 2), and let D0 and D1 denote the discs tangent to Y -axis and centered at S0 and S1 , respectively. We show that there is a family of parameters de ning T such that T : R 2 ! R 2 is a repelling system. Proposition 3. Suppose that the angles 0; 1 < 45 are non-commensurate with the full angle. Then for all x 62 D0 [ D1 , (2) dist(S0 ; T nx) ! 1 as n ! 1: Proof. The key idea in this is to study the rst-return map to P0 . We rst observe that an iterate of a point x 2 P0 ? D0 returning to P0 is always further away from S0 than the previous iterate leaving P0 (Figure 3). Let TP0 be the rst-return map to P0. We show that TP0 is well de ned for all x 2 P0 ? D0, and that p r22 r02 + 16 ? 8 2; (3) where r0 = dist(S0 ; x) and r2 = dist(S0; TP0 x). Proposition 3 then p follows from (3). Let r1 = dist(S1 ; Tx) (Figure 3) and let d = dist(S0; S1) = 2 2 for brevity. Using elementary geometry and the notation from Figure 3, we obtain: (4) r12 = d2 + r02 ? 2dr0 cos = d2 + r02 ? 2dr0 cos(135 ? ? ) (5) = d2 + r02 + 2dr0(cos(45 + ) cos ? sin(45 + ) sin ) (6) = d2 + r02 + 2dr0 cos(45 + ) cos ? 2d sin(45 + )dist(O; S0):
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Figure 3. Repelling system. (a) Single orbit. (b) Illustration to the proof of Theorem 3. The point x \is existing" from P0.
Since 0 < 45 , 0 < 45, < 90, from (6), we obtain (7) r12 d2 + r02 ? 2d:
Observe that Tx is not trapped in D1 since T0?1D1 \ P0 = (for otherwise 0 > 90). Hence, ?1 T x which \is exiting" from P . The we can repeat the above estimate for the iterate T?1 P0 1 corresponding to (7) inequality is (8) r22 d2 + r12 ? 2d: From (7) and (8), we nally obtain the estimate (3). The system described in Proposition 3 is a particular example of a class of repelling systems analyzed in [8]. Observe that the two xed cells: h00 : : : 0i and h11 : : : 1i are the only periodic cells in the above example. This follows from Proposition 3 as the orbit of a periodic cell must be bounded. Also in Example 1, h00 : : : 0i and h11 : : : 1i are the only periodic cells. However, in our nal example, apart from the two xed cells containing the centers of rotation, there are in nitely many other periodic cells. These cells form a self-similar con guration of periodic pentagons. The collection of these periodic cells almost completely tiles a global attractor for T (Figure 4 and 5).
Example 3. Attracting system with a self-similar structure of periodic cells on its attractor. Let us start from a simple geometric construction. Let A, B , and C be the
vertices of a triangle whose angles are 36, 72 and 72, respectively (Figure 4). Let S0 and S1 be the centers of the circles inscribed in in 4ABD and 4BCD, where D = L \ CA. Let us introduce the coordinate system so that the Y -axis be the line bisecting the angle \ABC , and such that S0 = (?1; 0). (The reader may check that then S1 = (k; ?(k + 1) tan 36), where k = 2 cos 72 is the similarity ratio between 4BCD and 4ABC .)
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The action of T restricted to 4ABC . The white regions in the right triangle are the interiors of \cells," maximal regions which follow the same pattern of visits to 4ABD and 4BCD. Figure 4.
The piecewise rotation T is de ned so that its restriction to the half-plane P0 = f(x; y) : x < 0g is the rotation about S0 by 0 = ?144 , and its restriction to the half-plane P1 = f(x; y) : x 0g is the rotation about S1 by 1 = 144. The triangle 4ABC plays a special role in the dynamics of T . Note that 4ABC is mapped under T into itself (Figure 4) and T j4ABC is almost 1-1: T (4ABC ) = T0(4ABD ? BD) [ T1 (4BCD) (9) = 4CAD0 ? AD0 [ 4CD0B = 4ABC ? AD0 [ D0; where D0 denotes a point symmetric to D with respect to the line S0S1 . One of the key properties of T is that the rst-return map to 4BCD of T j4ABC is essentially self-similar to T j4ABC . We shall use this idea in the proof of the following surprising result about the dynamical system T : R 2 ! R 2 . Theorem 1 (Periodicity). Let T be the piecewise rotation as de ned above. The triangle 4ABC splits into the union of an in nite number of periodic cells, the nonempty set R4ABC of non-eventually periodic points, and a set of zero Lebesgue measure. Theorem 2 (Global attractor). For every bounded set X 4ABC , there is some n > 0 such that T nX 4ABC . However, \ n T X = 4ABC: n>0
Remark 1. Even though the set R4ABC has Lebesgue measure zero, this set is \relatively
large" for its Hausdor dimension satis es 1 < dimH (R4ABC ) < 2. The Hausdor dimension is a nonnegative number which measures the \size" of a set. Two sets of Lebesgue measure zero may have di erent Hausdor dimensions. The reader may nd an elementary treatment of the Hausdor dimension in [4].
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Remark 2. Since the angles of rotation 0 and 1 are both commensurate with the full angle, every periodic cell consists of only periodic points. By Remark 1, with respect to Lebesgue measure, almost all points in 4ABC are periodic. Moreover, from Theorem 2, it follows that
the set R of non-eventually periodic points is contained in the inverse orbit of R4ABC , hence Corollary 1. With respect to Lebesgue measure, almost every point in R 2 is eventually periodic under T . Remark 3. The Poincare argument [7] shows that a non-eventually periodic cell in 4ABC must be a set of zero Lebesgue measure. Since all cells are convex, a non-eventually periodic cell must be either a line segment or a single point. Theorem 1 thus yields that the number of non-eventually periodic cells is uncountable. Remark 4. The orbits of some non-eventually periodic points never intersect the discontinuity line Y , for the backward orbit of Y is a countable union of line, and thus has Hausdor dimension one, and by Remark 1, the Hausdor dimension of non-eventually periodic points is larger than one. Proof of Theorem 1. We rst observe that the action of T on 4ABC is self-similar. Let E be the intersection of BC and the line bisecting the angle \CDB (Figure 4). Observe that on 4BED, the return time to 4BCD is one, thus the rst-return map T4BCD to 4BCD rotates 4BE 0D about S1 by 1 = 144. The triangle 4CDE returns back to 4BCD after three iterations, and in this process 4CDE is rotated by 1 = ? 0 = ?144 about the center S2 of the circle circumscribed in 4CDE . Hence, for x 2 4ABC ? Y , the maps T j4ABC and T4BCD are related by an ane order reversing similarity f1 whose similarity ratio k = 2 cos 72. Given a set U , let QU denote the collection of the interiors of periodic cells contained in U . The similarity f1 is then a bijection between Q4ABC and and its subset Q4BCD . Similarly, by applying the self-similarity argument once again, we conclude that there is a similarity f2 with the similarity ratio (k = 2 cos 72) which bijectively maps Q4BCD and Q4CDE . Let Q0 2 Q4ABC denote the xed open pentagon centered at S0 . From Figure 4, observe that Q4ABC splits into Q0 and three self-similar collections: Q4ABC = Q0 [ Q4BCD [ T1 Q4CDE [ T0 T1Q4CDE : (10) From (10), it immediately follows that Q4ABC is in nite, hence 4ABC contains an in nite number of periodic cells. It remains to be shown that the set of non-eventually periodic points in nonempty. This follows from Remark 1. Since the proof of Remark 1 involves tracking some details, here, we only list the important points and we leave the details to the reader. Let Z be the complement of all elements in Q4ABC . Using the self-similar structure of cells (10), one shows that the Housdor dimension s of the set Z satis es ks + 2(k2)s = 1, hence it is strictly between one and two. Then one argues that the collection W of eventually periodic points in Z is contained in a countable collection of lines, hence the Housdor dimension of R4ABC = Z ? W is also strictly between one and two.
Remark 4. Using the self-similar structure of periodic cells in 4ABC and 4BCD, it is not dicult to show that codings of periodic pentagons are given by the periodic blocks obtained from the substitutions: 0 ! 1 and 1 ! 100. More precisely, let F0 = f0g, and let Fn+1 be
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Figure 5. Illustration to the proof of Theorem 2. A number of single points denote the orbit of the point x.
Figure 6. Transition diagram. Every orbit of T must follow one of the allowable paths. The labeling of edges denotes the maximal number of iterations required for an orbit to enter a new region.
the set of blocks obtained by replacing 0 with 1 and replacing 1 withS 100 in the blocks from Fn. Then, all periodic codings are given by vwww : : : , where w 2 n0 Fn and v is a right subword of w. Proof of Theorem 2. The key idea of the proof is to construct a suitable partition of R 2 for which we can draw a transition diagram indicating that all orbits must be eventually trapped in 4ABC . One such partition is shown in Figure 5, and its transition diagram is shown in Figure 6.
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An important observation used to produce the transition diagram is that iterates of T , when restricted to these various partitions, become simple translations. For example, T 2jP01up acts ?! as a translation by AD0. From Diagram 4, we see that any orbit is eventually trapped in 4ABC , or it is trapped in H = P01up [ P10up ? 4ABC . However, no orbit?!can remain in H , as we now show. ?! Observe (Figure 4) that H fv AD +w D0A; v 0; w > 0g. Moreover, since T jP10up P1, ?! ?! T 2jP01up is the translation by AD0 and since T 2jP10up is the translation by DA, for all x 2 H , ?! ?! T 2x decreases one of the coordinates v or w by 1, where x = v AD +w D0A. Thus the time which the orbit of x 2 H may spend in H is bounded by (v + w + 2). From Diagram 4, we nally conclude that every orbit is eventually trapped in 4ABC . Moreover, there is some n such that T nX 4ABC for every bounded set X 4ABC , which is the rst conclusion of Theorem 2. The second conclusion of Theorem 2 follows from (9) for T j4ABC does not decrease Lebesgue measure.
Remark 6. Example 2 illustrates the existence of a speci c piecewise rotation for which
there are in nitely many periodic cells of positive measure. Clearly, the above example is very special. It is interesting to ask: Question. How many are there examples of piecewise rotations with an in nite number of periodic cells? It turns out that there is an uncountable number of non-isomorphic systems with in nitely many periodic cells which are discs. Such systems are obtained by using the return-map techniques and by \suitably perturbing" the attracting system described in Example 1. However, details describing this construction are rather complex [8]. We do not know whether systems with in nite number of periodic cells are generic. In conclusion, we would like to invite the reader to exlore other examples of piecewise rotations. For instance, we propose that the reader study the following example of an invertible piecewise rotation T . Proposition 4. Let T : R 2 ! R 2 have the centers of rotation S0 = (?0:5; ?0:5) and S1 = (:5; :5) and angles of rotation 0 = 1 == 90. Then all points in the plane are periodic under T . The structure of cells essentially coincides with the integer lattice, and there exist cells of periods larger than any arbitrary integer. Even though the choice of the parameters from Proposition 4 is very special, we conjecture that for all invertible piecewise rotations T , almost all points are periodic. We note that this conjecture cannot be improved for all invertible T . For if the angles of rotations are noncommensurate with the full angle, the periodic cells are either single points or they are discs. Since the union of nonoverlapping discs and countable single points cannot cover the plane, it follows that for these T , there exist nonperiodic cells. Acknowledgments. We found Example 2 by experimenting with a computer. These experiments were inspired by Bruce Kitchens after his lecture on piecewise ane maps of the torus. In the context of piecewise ane maps, Adler, Kitchens, and Tresser analyzed in [1]
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in great detail other two examples of a an invertible piecewise rotation with a self-similar periodic structure. In the proof of Theorem 1 we used some techniques from [1]. The idea of an expository article on the subject of piecewise rotations was warmly encouraged by Bob Devaney. I would like to express my gratitude to Bob for his interest and continual support in composing this article. References
[1] Roy Adler, Bruce Kitchens, and Charles Tresser. Dynamics of piecewise ane maps of the torus. Watson Research Center, preprint, 1998. [2] Michael Boshernitzan and Isaac Kornfeld. Interval translation mappings. Ergodic Theory and Dynamical Systems, 15:821{831, 1995. [3] Robert L. Devaney. An introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading MA, 1989. [4] K. Falconer. Fractal Geometry. John Wiley, Chichester, 1990. [5] Arek Goetz. So c subshifts and piecewise isometric systems. Ergodic Theory and Dynamical Systems (to appear). [6] Arek Goetz. Dynamics of Piecewise Isometries. PhD thesis, University of Illinois at Chicago, 1996. [7] Arek Goetz. Dynamics of a piecewise rotation. Continuous and Discrete Dynamical Systems, 4(4):593{608, 1998. [8] Arek Goetz. Perturbations of 8-attractors and births of satellite systems. International Journal of Bifurcation and Chaos, 8:1937{1956, 10 1998. [9] Eugene Gutkin and Nicolai Haydn. Topological entropy of generalized polygon exchanges. Ergodic Theory and Dynamical Systems, 17:849{867, 1997. [10] Hans Haller. Rectangle exchange transformations. Manatshefte fur Mathematik, 3:215 {232, 1981. [11] Anatole Katok and Boris Hasselblatt. Introduction to the modern theory in Dynamical Systems. Cambridge University Press, 1995. [12] Michael Keane. Interval exchange transformations. Mathematische Zeitung, 141:25{31, 1975. [13] Gilbert Levitt. La dynamique des pseudogroupes de rotations. Invent. Math., 113:633{670, 1993. [14] Maciej J. Ogorzalek. Complex behavior in digital lters. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2(1):11{29, 1992. [15] Clark Robinson. Dynamical Sytems, Stability, Symbolic Dynamics and Chaos. CRC Press, 1995. [16] Tabashnikov. Panorames et syntheses. Soc. Math. France, 1995. Department of Mathematics, Boston University, 111 Cummington St., Boston, Ma. 02215
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