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Notes on Piecewise Isometries NOTES ON PIECEWISE ISOMETRIES AREK GOETZ Abstract. This document contains selected notes used in the third part of the Dynamical Systems. They will evolve and will be updated after almost every lecture. Date: May 17, 2003. 1991 Mathematics Subject Classi¯cation. Primary 58F03; Secondary 22C05. 1 2 AREK GOETZ 1. Rotations De¯nition 1. A (planar) isometry is a map R : C ! C which preserves distances, that is such that for all z1;z2 2 C; jz1 ¡ z2j = jRz1 ¡ Rz2j. All non-identity planar isometries can be divided into categories according to the number of ¯xed points that they have. Maps with no ¯xed points are translations, those with exactly one ¯xed points are rotations and those with more than one ¯xed point are reflections. As the following propositions state, all planar piecewise isometries are conveniently de- scribed using complex notation. Proposition 1. Any rotation or translation can be written as R(z)=½z + t where ½ = eI®, 0 · ®<2¼. The map R ¯rst rotates a point z by angle ®, and then it translates the result ½z by t.If ½ = 1 that is when ® = 0, the map R is a translation. Otherwise, if ½ =6 1, the map R,a proper rotation, has exactly one ¯xed point, z0 = t=(1 ¡ ½). Proposition 2. Any reflection can be written as R(z)=½z where ½ = eI®, 0 · ®<2¼ and z denotes a conjugation of z. The conjugation z flips a point z with respect the real axis. Basic properties of the dynamical systems of rotations are well understood. There are two distinct cases, (a) rational rotations, and (b) irrational rotations. A rotation R is rational if ®=¼ 2 Q. Otherwise we say that R is irrational. Iterates of all points under rational rotations are periodic. Orbits of all points other than the ¯xed point are in¯nite under irrational rotations. These orbits are dense in a circle. Homework Problems Exercise 1. Show that the map R : C ! C, R(z)=½z + t where ½; t 2 C, j½j = 1 is an isometry. Exercise 2. Show that the map R : C ! C, R(z)=½z, j½j = 1 is an isometry. Exercise 3. Show that for an irrational rotation R, all z 2 C either z is ¯xed or R or the orbit of z is in¯nite. R ® ¼ k : C Exercise 4. Let be a rotation by the angle =2 m Show that All points in are periodic of period m. Show that ½ = ei® satis¯es ½m = 1, that is ½ is a root of unity. Exercise 5. Show that under a proper rotation R the orbit of any point lies on a circle centered at the ¯xed point z0 = t=(1 ¡ ½)ofR. Exercise 6. Show that a rotation R(z)=½z + t,(j½j6= 1) is conjugate to a rotation 0 R (z)=½z via the translation h(z)=z ¡ z0, C ¡¡¡R! C ? ? ? ? hy yh 0 C ¡¡¡R ! C; where z0 = t=(1 ¡ ½) is the ¯xed point for R. Exercise 7. Show that for the map R(z)=½z + t, the set of ¯xed points is the a line L with slope 1=2 arg(½). Then show that R(z) reflects z with respect to L. NOTES ON PIECEWISE ISOMETRIES 3 Exercise 8. Show that if an orbit of a point z under an isometry R is unbounded, that is, if for all M, there is an iterate T k(z), jT k(z)j >M, then R must be a translation. Exercise 9. Prove that an isometry is a bijection (1-1 and onto). i¼ Computer Exercise 10. Graph the ¯rst 100 iterates of 1 under the rotation R(z)=e k where k = 3+1=2. Then graph the ¯rst 100 iterates of 1 under the rotation R(z) for k 1 =3+2+1=3 . What can you say about the orbit behavior in each case. 2. Piecewise Rotations In the previous section, we saw that a Euclidean rotation T in the plane has a rela- tively simple dynamics. For every point x other than the center of rotation T the orbit fx; T x; T (Tx);:::g is a ¯nite set or it is a dense subset of a circle when T . This depends on whether T is a rotation by an angle commensurate with the full angle. Composing two or more rotations in the plane leads to complicated and fascinating dynam- ical systems. These systems are also surprising, since they can have self-similar structures, attractors, repellers, which are usually observed in systems whose generating map has either contacting or expanding properties. The dynamics of maps changes dramatically if we introduce discontinuities in the gen- erating maps. In this section we will introduce a class of maps which are rotations when restricted to subsets of the domain. De¯nition 2. A map T : X ! X is called a piecewise rotation with atoms P = fPjgj2N if Tx = ½jx + tj 2 Pj for some complex numbers: sj and ½j such that j½jj = 1 for all j 2 N, and ½j =6 1 for at least one j. If all ½j are roots of unity, then T is called a rational piecewise rotation. Frequently, we will refer to sets Pi as atoms. Figure 1. An example of a piecewise rotation with two atoms. The angle ® = ¼=5. The right ¯gure represents a partition of space 4oca into sets that follow the same pattern of visits to the atoms. Iterates of each colored pentagon never get broken apart by the discontinuity. In Figure 1, we illustrated one of the most simple examples of a piecewise rotation T de¯ned on a triangle X = 4oac. Associated with this map are two subdividing X triangles, P0 and P1 on which T acts as a rotation. On P0, T rotates P0 by ¼ ¡ ¼=5 about S0.On P1, T is a rotation by ¼=5 about S1. The center S0 is the center of the circle subscribed in P0 and S1 is the center of the circle circumscribed in P1. The right ¯gure is a partition of triangle X into sets that follow the same periodic pattern of visits to the atoms P0 and P1. It is surprising that this mosaic appears to be fractal. In order to study it we now de¯ne a convenient and widely used in dynamical system tool called symbolic encoding. 4 AREK GOETZ De¯nition 3. Coding. The partition P of X associated to a piecewise rotation T gives rise to a natural one-sided coding map i : X ! ­=N N for T . The map i encodes the forward orbit of a point by recording the indices of atoms visited by the orbit, that is i(x)=w0w1 :::, T jx 2 P where wj . De¯nition 4. Cells. A set of all points following the same coding ! 2 ­ under T will be called a cell. All cells and are convex provided that the atoms fPig are convex (Goetz 2000). Piecewise rotations are natural two dimensional generalizations of well studied interval exchange transformations studied in (Arnoux, Ornstein & Weiss 1985, Boshernitzan 1988, Boshernitzan 1985, Katok 1980, Masur 1982, Keane 1977, Keynes & Newton 1976, Veech 1987, Veech 1982, Veech 1978) and interval translation maps (Boshernitzan & Kornfeld 1995, Troubetzkoy & Schmeling 2000). Invertible piecewise rotations (Adler, Kitchens & Tresser 2001, Ashwin & Fu 2001, Ashwin & Fu 2002, Haller 1981, Gutkin & Haydn 1997, Kahgng 2002, Poggiaspalla 2002) (those that preserve Lebesgue measure) are closely related to the theory of dual billiards (Tabachnikov 1995) and Hamiltonian systems (Scott, Holmes & Milburn 2001). A somewhat unusual application is found outside of mathematics, in electrical engineer- ing, in particular in the theory of digital ¯lters (Ashwin 1996, Ashwin, Chambers & Petrov 1997, Chua & Lin 1988, Chua & Lin 1990, Davies 1992, Kocarev, Wu & Chua 1996, OgorzaÃlek 1992). Digital ¯lters are algorithms widely incorporated in electronic components in con- temporary electronics devices such as cellular phones, radio devices and voice and image recognition systems. The most basic examples of piecewise isometries are exchanges of intervals (Arnoux et al. 1985, Boshernitzan 1988, Boshernitzan 1985, Katok 1980, Masur 1982, Keane 1977, Keynes & Newton 1976, Veech 1987, Veech 1982, Veech 1978). De¯nition 2 (Interval Exchange Transformation.) A piecewise isometry T : X ! X is called a interval exchange transformation with atoms if X is an left open interval, T is Lebesgue measure preserving, T restricted to each atom is a translation. Homework Problems Exercise 11. Determine all ¯xed points for the map T illustrated in Figure 1. Exercise 12. Show that there are no periodic points of period 2 for the map T illustrated in Figure 1. Exercise 13. Find a periodic point of period 3 for the map T illustrated in Figure 1. ½ 2¼ i 2¼ j 2f ; ;:::; g j¼ i j¼ Exercise 14. Let = cos 5 + sin 5 . For each 1 3 9 , write down cos 5 + sin 5 as a monomial expression in ½. ½ 2¼ i 2¼ Exercise 15. Let = cos 5 + sin 5 be the ¯rst ¯fth root of unity. In Figure 1 let 3 the common vertices of the triangles P0 and P1 be o =0,b = ¡1. Show that a = ½ 2 3 and c = ½ + ½ + ½ . Then show that the map T rotates triangle P0 by (¼ ¡ ¼=5) and then 2 3 translates it by c = ½+½ +½ . Finally show that the triangle P1 is rotated by (¼=5¡¼) under T , and then it is translated by a = ½3 (Hint: Exercise 14 and also 1 + ½ + ½2 + ½3 + ½4 = 0). *Exercise 16. Show that for the map T illustrated in Figure 1, the centers of rotations S = ¡ ½ ¡ ½2 ½3 S = ½ ½2 ½3 ½ 2¼ i 2¼ 0 =1 5( 2+ +2 ) and 1 =1 5(1 + 2 +3 +4 ) where = cos 5 + sin 5 .
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