spatiotemporal cats or, try herding 6 cats siminos/spatiotemp, rev. 6798: last edit by Predrag Cvitanovi´c,03/08/2019 Predrag Cvitanovi´c,Boris Gutkin, Li Han, Rana Jafari, Han Liang, and Adrien K. Saremi April 1, 2019 Contents 1 Cat map7 1.1 Adler-Weiss partition of the Thom-Arnol’d cat map.......7 1.2 Adler-Weiss partition of the Percival-Vivaldi cat map......9 1.2.1 Adler-Weiss linear code partition of the phase space... 16 1.3 Perron-Frobenius operators and periodic orbits theory of cat maps 18 1.3.1 Cat map topological zeta function............. 19 1.3.2 Adler / Adler98........................ 20 1.3.3 Percival and Vivaldi / PerViv................ 21 1.3.4 Isola / Isola90......................... 22 1.3.5 Creagh / Creagh94...................... 23 1.3.6 Keating / Keating91..................... 24 1.4 Green’s function for 1-dimensional lattice............. 25 1.5 Green’s blog.............................. 28 1.6 Z2 = D1 factorization......................... 31 1.7 Any piecewise linear map has “linear code”............ 32 1.8 Cat map blog............................. 33 References.................................. 47 1.9 Examples................................ 53 exercises 61 2 Statistical mechanics applications 67 2.1 Cat map................................ 67 2.2 New example: Arnol’d cat map................... 67 2.3 Diffusion in Hamiltonian sawtooth and cat maps......... 70 References.................................. 75 3 Spatiotemporal cat 77 3.1 Elastodynamic equilibria of 2D solids............... 79 3.2 Cats’ GHJSC16blog.......................... 80 References.................................. 94 4 Ising model in 2D 97 4.1 Ihara zeta functions.......................... 101 4.1.1 Clair / Clair14......................... 101 2 CONTENTS 4.1.2 Maillard............................ 105 4.2 Zeta functions in d=2......................... 108 References.................................. 111 5 Checkerboard 117 5.1 Checkerboard model......................... 117 5.1.1 Checkerboard literature................... 118 References.................................. 119 6 Hill’s formula 120 6.1 Generating functions; action..................... 120 6.2 Homoclinic and periodic orbit actions in chaotic systems.... 122 6.3 Hill’s formula............................. 124 6.3.1 Generating function literature................ 130 6.4 Noether’s theorem.......................... 132 References.................................. 132 7 Symbolic dynamics: a glossary 135 7.1 Symbolic dynamics, inserts..................... 139 References.................................. 139 8 Han’s blog 141 8.1 Rhomboid corner partition...................... 143 8.2 Reduction to the reciprocal lattice.................. 157 8.3 Rhomboid center partition...................... 163 8.3.1 Reduction to the reciprocal lattice............. 173 8.4 Time reversal............................. 175 8.5 Reduction to the fundamental domain............... 182 8.6 Spatiotemporal cat partition..................... 185 8.7 Running blog............................. 194 References.................................. 241 9 Frequencies of Cat Map Winding Numbers 244 9.1 Introduction.............................. 244 9.2 Numbers of periodic orbits..................... 245 9.2.1 Keating’s counting of periodic points........... 246 9.3 Relative frequencies of cat map words............... 246 9.3.1 Numerical computations.................. 247 9.3.2 Entropy............................ 248 9.4 Spatiotemporal cat.......................... 249 9.4.1 Frequencies of symbols................... 250 9.4.2 Frequency of blocks of length two............. 251 9.4.3 Frequencies of [2×2] spatiotemporal domains...... 253 9.5 Summary................................ 253 References.................................. 254 6814 (predrag–6798) 3 03/08/2019 siminos/spatiotemp CONTENTS 10 Symbolic Dynamics for Coupled Cat Maps 255 10.1 Introduction.............................. 255 10.2 Arnol’d Cat Map........................... 256 10.2.1 Periodic orbits - first approach............... 257 10.2.2 Periodic orbits - second approach............. 257 10.2.3 Symbolic dynamics for 1 particle.............. 260 10.3 Coupled Cat Maps.......................... 262 10.3.1 Computation of symbols................... 262 10.3.2 Nature of our alphabet.................... 264 10.3.3 Admissibility of sequences of more than 2 symbols... 266 10.4 Future work................................ 268 References.................................. 269 11 Rana’s blog 272 References.................................. 274 12 Adrien’s blog 275 12.1 Introduction.............................. 275 12.2 Arnol’d cat map............................ 276 12.2.1 First approach......................... 276 12.2.2 2nd approach......................... 277 12.2.3 Computing periodic orbits................. 277 12.3 Coupled Cat Maps.......................... 286 12.3.1 Periodic orbits......................... 286 12.3.2 Ergodic orbits......................... 287 12.4 Adrien’s blog............................. 291 References.................................. 294 13 Spatiotemporal cat, blogged 296 References.................................. 320 03/08/2019 siminos/spatiotemp 4 6814 (predrag–6798) CONTENTS This is a project of many movable parts, so here is a guide where to blog specific topics (or where to find them) • chapter 1.9 Cat map • sect. 6.1 1-dimensional action • sect. 1.5 1-dimensional / cat map lattice Green’s functions • sect. 1.8 cat map blog • chapter3 d-dimensional spatiotemporal cat • chapter 13 Spatiotemporal cat blog • chapter 8.2 Reduction to the reciprocal lattice • sect. 3.2 Herding five cats [1] edits • chapter4 2D Ising model; Ihara and multi-dimensional zetas • chapter5 Checkerboard • chapter6 Hill’s formula • chapter7 Symbolic dynamics: a glossary • chapter8 Han’s blog • chapter 11 Rana’s blog • chapter 12 Adrien’s blog Blog fearlessly: this is your own lab-book, a chronology of your learning and research that you might find invaluable years hence. 6814 (predrag–6798) 5 03/08/2019 siminos/spatiotemp Chapter 1 Cat map If space is infinite, we are in no particular point in space. If time is infinite, we are in no particular point in time. — The Book of Sand, by Jorge Luis Borges What is a natural way to cover the torus, in such a way that the dynamics and the partition borders are correctly aligned? You are allowed to coordinatize the unit torus by any set of coordinates that covers the torus by a unit area. The origin is fixed under the action of A, and straight lines map into the straight lines, so Adler and Weiss did the natural thing, and used parallelograms (fol- lowing Bowen [28] we shall refer to such parallelograms as ‘rectangles’) with edges parallel to the two eigenvectors of A. Adler and Weiss observed that the torus in the new eigen-coordinates is covered by two rectangles, labelled A and B in figure 1.1. 1 1.1 Adler-Weiss partition of the Thom-Arnol’d cat map Figure 1.1 for the canonical Thom-Arnol’d cat map remark 1.4 2 1 A = : (1.1) 1 1 Note that [5] 2 2 1 1 1 = ; (1.2) 1 1 1 0 so each of equivalence classes with respect to centralizer is split into two equiv- alence classes with respect to the group {±An j n 2 Zg. (See also (4.18).) 1Predrag 2018-02-09: (1) motivate Manning multiples by doing the 1D circle map first. Maybe Robinson [76] does that. (2) motivate spatiotemporal cat by recent Gutkin et al. many-body paper 7 CHAPTER 1. CAT MAP B u W (0, 1) A u − s W ( 1, 0) W (3, 2) u W (0, 0) s s s s s (a) W (0, 0) W (0, 1) W (1, 1) W (2, 1) W (3, 1) 3' f ( ) 2' 1' 1 W u (0, 1) f ( ) u W (− 1, 0) W s (3, 2) W u (0, 0) s W (2, 2) (b) W s (0, 0) W s (0, 1) W s (1, 1) W s (2, 1) f(3) 5 4 f(5) 3 f(1) 1 f(4) 0 f(0) 4 3 5 B A 0 (c) 1 Figure 1.1: (a) Two-rectangles Adler-Weiss generating partition for the canon- ical Arnol’d cat map (1.1), with borders given by stable-unstable manifolds of the unfolded cat map lattice points near to the origin. (b) The first iterate of the partition. (c) The iterate pulled back into the generating partition, and the corresponding 5-letter transition graph. In (b) and (c) I have not bothered to re- label Crutchfield partition labels with our shift code. This is a “linear code,” in the sense that for each square on can count how many side-lengths are needed to pull the overhanging part of f(x) back into the two defining squares. (Figure by Crutchfield [34]) 03/08/2019 siminos/spatiotemp 8 6814 (predrag–6798) CHAPTER 1. CAT MAP String people, arXiv:1608.07845, find the identity 2 1 1 1 1 0 = = LL> (1.3) 1 1 0 1 1 1 significant: “The map corresponds to successive kicks, forwards and back- wards along the light cone [...]” As another example, with s = 4, Manning [67] discusses a Markov partition for the cat map (also discussed by Anosov, Klimenko and Kolutsky [5]) 3 1 A = : (1.4) 2 1 2CB In order to count all admissible walks, one associates with the transition graph such as the one in figure 1.1 (c) the connectivity matrix 1 1 C = ; (1.5) 1 2 where Cij is the number of ways (number of links) of getting to i from j. 1.2 Adler-Weiss partition of the Percival-Vivaldi cat map As illustrated in figure 1.2, the action of the cat map in the Percival-Vivaldi [73] “two-configuration representation” is given by the antisymmetric area preserv- ing [2×2] matrix 0 1 A = (1.6) −1 s For the Arnol’d value s = 3, in one time step the map stretches the unit square into a parallelogram, and than wraps it around the torus 3 times, as in fig- ure 1.2. Visualise the phase space as a bagel, with x0 axis a circle on the outside of the bagel. This circle is divided into three color segments, which map onto each other as you got in the x1 axis direction. Now apply the inverse map - you get 3 strips intersecting the the above strips, for 9 rectangles in all: a full shift, i.e., a ternary Smale horseshoe. So on the torus there are only 3 strips - there is no distinction between the two outer letters A1 = {−1; 2g = {red, yellow}, it is the same third strip.